Alexey P. Stakhov - The Mathematics of Harmony From Euclid to Contemporary Mathematics and Computer Science

This page intentionally left blank
World Scientific
N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Stakhov, A. P. (Alexey Petrovich)
The mathematics of harmony : from Euclid to contemporary mathematics and computer science / by Alexey
Stakhov ; assisted by Scott Olsen.
p. cm. -- (Series on knots and everything ; v. 22)
Includes bibliographical references and index.
ISBN-13: 978-981-277-582-5 (hardcover : alk. paper)
ISBN-10: 981-277-582-X (hardcover : alk. paper)
1. Fibonacci numbers. 2. Golden section. 3. Mathematics--History. 4. Computer science. I. Olsen, Scott
Anthony II. Title.
QA246.5.S73 2009
512.7'2--dc22
2009021159
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or
mechanical, including photocopying, recording or any information storage and retrieval system now known or to
be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center,
Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from
the publisher.
Printed in Singapore.
EH - The Mathematics Harmony.pmd
1
7/17/2009, 9:41 AM
Dedication
v
With deep gratitude to my parents, my darling father, Peter Stakhov,
who was killed in 1941 during World War II (19411945),
and to my darling mother, Daria Stakh, who passed away in 2001,
and to my darling teacher Professor Alexander Volkov,
who passed away in 2007.
This page intentionally left blank
Preface
vii
Preface
Academician Mitropolsky’s commentary on the scientific
research of Ukrainian scientist Professor Alexey Stakhov,
Doctor of Engineering Sciences
I have followed the scientific career of Professor Stakhov for a long time—
seemingly since the publication of his first book, Introduction into Algorithmic
Measurement Theory (1977), which was presented by Professor Stakhov in
1979 at the scientific seminar of the Mathematics Institute of the Ukrainian
Academy of Sciences. I became especially interested in Stakhov’s scientific
research after listening to his brilliant speech at a session of the Presidium of
the Ukrainian Academy of Sciences in 1989. In his speech, Professor Stakhov
reported on scientific and engineering developments in the field of “Fibonacci
computers” that were conducted under his scientific supervision at Vinnitsa
Technical University.
I am very familiar with Stakhov’s scientific works as many of his papers
were published in various Ukrainian academic journals at my recommenda
tion. In April 1998, I invited Professor Stakhov to report on his scientific re
search at a meeting of the Ukrainian Mathematical Society. His lecture pro
duced a positive reaction from the members of the society. At the request of
Professor Stakhov, I wrote the introduction to his book, Hyperbolic Fibonacci
and Lucas Functions, which was published in 2003 in small edition. In recent
years, I have been actively corresponding with Professor Stakhov, and we have
discussed many new scientific ideas. During these discussions I became very
impressed with his qualifications and extensive knowledge in regard to his
research in various areas of modern science. In particular, I am impressed by
his knowledge in the field of mathematics history.
The main feature of Stakhov’s scientific creativity consists of his uncon
ventional outlook upon ancient mathematical problems. As an example, I shall
begin with my review of his book Introduction into Algorithmic Measurement
Theory (1977). This publication rewarded Professor Stakhov with recogni
tion in the field of modern theoretical metrology. In this book, Professor Sta
khov introduced a new mathematical direction in measurement theory—the
Algorithmic Measurement Theory.
Alexey Stakhov
MATHEMATICS OF HARMONY
viii
In 1993, I recommended a publication of an innovative paper, prepared by
Professor Alexey Stakhov and Ivan Tkachenko, entitled “Fibonacci Hyper
bolic Trigonometry,” for publication in the journal Reports of the Ukrainian
Academy of Sciences. The paper addressed a new theory of hyperbolic Fibonacci
and Lucas functions. This paper demonstrated the uniqueness of Stakhov’s
scientific thinking. In fact, the classical hyperbolic functions were widely
known and were used as a basis of nonEuclidean geometry developed by Ni
kolay Lobachevsky. It is quite peculiar that at the end of 20th century Ukrai
nian scientists Stakhov and Tkachenko discovered a new class of the hyper
bolic functions based on the Golden Section, Fibonacci and Lucas numbers
that has “strategic” importance for the development of modern mathematics
and theoretical physics.
In 1999, I also recommended Stakhov’s article “A Generalization of the
Fibonacci QMatrix”—which was presented by the author in English—to be
published in the journal Reports of the Ukrainian Academy of Sciences (1999,
Vol. 9). In this article, Professor Stakhov generalized and developed a new
theory of the Qmatrix which had been introduced by the American mathe
matician Verner Hoggatt—a founder of the FibonacciAssociation. Stakhov
introduced a concept of the Qpmatrices (p=0, 1, 2, 3...), which are a new class
of square matrices (a number of such matrices is infinite). These matrices are
based on socalled Fibonacci pnumbers, which had been discovered by Sta
khov while investigating “diagonal sums” of the Pascal triangle. Stakhov dis
covered a number of quite unusual properties of the Qpmatrices. In particu
lar, he proved that the determinant of the Qpmatrix or any power of that
matrix is equal to +1 or 1. It is my firm belief that a theory of Qpmatrices
could be recognized as a new fundamental result in the classic matrix theory.
In 2004, The Ukrainian Mathematical Journal (Vol. 8), published Stakhov’s
article “The Generalized Golden Sections and a New Approach to Geometri
cal Definition of Number.” In this article, Professor Stakhov obtained mathe
matical results in number theory. The following are worth mentioning:
1. A Generalization of the Golden Section Problem. The essence of
this generalization is extremely simple. Let us set a nonnegative inte
ger (p=0, 1, 2, 3, ...) and divide a line segment АВ at the point C in the
following proportion:
CB  AB 
=
AC  CB 
p
Preface
ix
We then get the following algebraic equation:
xp+1 = xp + 1.
The positive roots of this algebraic equation were named the General
ized Golden Proportions or the Golden pproportions tp. Let’s ponder upon
this result. Within several millennia, since Pythagoras and Plato, man
kind widely used the known classical Golden Proportion as some unique
number. And at the end of the 20th century, the Ukrainian scientist
Stakhov has generalized this result and proved the existence of the in
finite number of the Golden Proportions; as all of them have the same
right to express Harmony, as well as the classical Golden Proportion.
Moreover, Stakhov proved that the golden pproportions τp (1≤τp≤2)
represented a new class of irrational numbers, which express some un
known mathematical properties of the Pascal triangle. Undoubtedly,
such mathematical result has fundamental importance for the develop
ment of modern science and mathematics.
2. Codes of the Golden рproportions. Using a concept of the golden
рproportion, Stakhov introduced a new definition of real number in
the form:
A=
∑ a τ , (a ∈ {0,1})
i
i p
i
i
He named this sum the “Code of the golden рproportion.” Stakhov proved
that this concept, which is an expansion of the wellknown Newton’s definition
of real number, could be used for the creation of a new theory for real numbers.
Furthermore, he proved that this result could also be used for the creation of
new computer arithmetic and new computers—Fibonacci computers. Stakhov
not only introduced the idea of Fibonacci computers, but he also organized the
engineering projects on the creation of such computer prototypes in the Vinnit
sa Polytechnic Institute from 19771995. 65 foreign patents for inventions in
the field of Fibonacci computers have been issued by the state patent offices of
the United States, Japan, England, France, Germany, Canada, and other coun
tries; these patents confirmed the significance of Ukrainian science and of Pro
fessor Stakhov’s work in this important computer area.
In recent years, the area of Professor Stakhov’s scientific interests has moved
more and more towards the area of mathematics. For example, his lecture “The
Golden Section and Modern Harmony Mathematics” delivered at the Seventh
International Conference on Fibonacci Numbers and their Applications in Graz,
Austria in 1996, and then repeated in 1998 at the Ukrainian Mathematical So
ciety, established a new trend in Stakhov’s scientific research. This lecture was
impressive and it created wide discussion on Stakhov’s new research.
Alexey Stakhov
MATHEMATICS OF HARMONY
x
Currently, Professor Stakhov is an actively working scientist who publishes
his scientific papers in many internationally recognized journals. Most recently,
he has published many fundamental papers in the international journals: Com
puters & Mathematics with Applications; The Computer Journal; Chaos, Solitons
& Fractals; Visual Mathematics; and others. This fact demonstrates, undoubt
edly, tremendous success not only for Professor Stakhov, but also for Ukrainian
science.
Stakhov’s articles are closing a cycle of his longterm research on the
creation of a new direction in mathematics: Mathematics of Harmony. One
may wonder what place in the general theory of mathematics this work may
have. It seems to me that in the last few centuries as Nikolay Lobachevsky
said, “Mathematicians have turned all their attention to the advanced parts
of analytics, and have neglected the origins of Mathematics, and are not
willing to dig the field that has already been harvested by them and left
behind.” As a result, this has created a gap between “Elementary Mathemat
ics”—the basis of modern mathematical education—and “Advanced Mathe
matics.” In my opinion, the Mathematics of Harmony developed by Profes
sor Stakhov fills that gap. Mathematics of Harmony is a huge theoretical
contribution to the development of “Elementary Mathematics,” and as such
should be considered of great importance for mathematical education.
It is imperative to mention that Professor Stakhov focuses his organiza
tional work on stimulating research in the field of theory surrounding Fibonacci
numbers and the Golden Section; he also assists in spreading knowledge among
broad audiences inside the scientific community. In 2003, under Professor
Stakhov’s initiative and scientific supervision, the international conference
on “Problems of Harmony, Symmetry, and the Golden Section in Nature, Sci
ence, and Art” was held. At this conference, Professor Stakhov was elected as
President of the International Club of the Golden Section, confirming his of
ficial status as leader of a new scientific direction that is actively progressing
the modern science.
Professor Stakhov proposed the discipline “Mathematics of Harmony and
the Golden Section” for the mathematical faculties of pedagogical universities.
In essence, this mathematical discipline can be considered the beginning of
mathematical education reform—which is based on the principles of Harmony
and the Golden Section. It should be noted that such discipline was delivered
by Professor Stakhov during 20012002 for the students and faculty of physics
and mathematics at Vinnitsa State Pedagogical University. I have no doubts
about the usefulness of such discipline for future teachers in mathematics and
physics. I believe that Professor Stakhov has the potential to write a textbook
Preface
xi
on this discipline for pedagogical universities, and also a textbook on Mathematics
of the Golden Section for secondary schools.
It is clear to me that “Mathematics of Harmony,” created by Professor
Stakhov, has huge interdisciplinary importance as this mathematical disci
pline touches the bases of many sciences, including: mathematics, theoretical
physics, and computer science. Stakhov suggested mathematical education
reform based on the ideas of Harmony and the Golden Section. This reform
opens the doors for the development of mathematical and general education
curriculum. It would greatly contribute to the development of the new scien
tific outlook based on the principles of Harmony and the Golden Section.
Yuri Mitropolsky
Doctor of Sciences in Theoretical Mechanics, Professor
Academician of the National Academy of Sciences of Ukraine
Academician of the Russian Academy of Sciences
Honorable Professor: The Mathematics Institute of the National
Academy of Sciences of Ukraine
EditorinChief of the Ukrainian Mathematical Journal
This page intentionally left blank
Contents
xiii
Contents
Introduction: Three “Key” Problems of Mathematics on the Stage
of its Origin
xix
Acknowledgements
xxxix
Credits and Sources
xIiii
Part I. Classical Golden Mean, Fibonacci
Numbers, and Platonic Solids
Chapter 1. The Golden Section
1.1.
1.2.
1.3.
1.4.
1.5.
1.6.
1.7.
1.8.
1.9.
1.10.
1.11.
1.12.
1.13.
1.14.
1.15.
Geometric Definition of the Golden Section
Algebraic Properties of the Golden Mean
The Algebraic Equation of the Golden Mean
The Golden Rectangles and the Golden Brick
Decagon: Connection of the Golden Mean to the Number π
The Golden Right Triangle and the Golden Ellipse
The Golden Isosceles Triangles and Pentagon
The Golden Section and the Mysteries of Egyptian Culture
The Golden Section in Greek Culture
The Golden Section in Renaissance Art
De Divina Proportione by Luca Pacioli
A Proportional Scheme of the Golden Section in Architecture
The Golden Section in the Art of 19th and 20th Centuries
A Formula of Beauty
Conclusion
2
7
11
20
24
25
27
32
37
41
46
51
53
56
59
Chapter 2. Fibonacci and Lucas Numbers
2.1.
2.2.
2.3.
2.4.
2.5.
Who was Fibonacci?
Fibonacci’s Rabbits
Numerology and Fibonacci Numbers
Variations on Fibonacci Theme
Lucas Numbers
60
62
67
72
77
Alexey Stakhov
MATHEMATICS OF HARMONY
xiv
2.6. Cassini Formula
83
2.7. Pythagorean Triangles and Fibonacci Numbers
85
2.8. Binet Formulas
90
2.9. Fibonacci Rectangle and Fibonacci Spiral
95
2.10. Chemistry by Fibonacci
97
2.11. Symmetry of Nature and the Nature of Symmetry
100
2.12. Omnipresent Phyllotaxis
105
2.13. “Fibonacci Resonances” of the Genetic Code
110
2.14. The Golden Section and Fibonacci Numbers in Music and Cinema 112
2.15. The Music of Poetry
116
2.16. The Problem of Choice: Will Buridan’s Donkey Die?
120
2.17. Elliott Waves
125
2.18. The Outstanding Fibonacci Mathematicians of the 20th Century 129
2.19. Slavic “Golden” Group
132
2.20. Conclusion
136
Chapter 3. Regular Polyhedrons
3.1.
3.2.
3.3.
3.4.
3.5.
3.6.
Platonic Solids
Archimedean Solids and Starshaped Regular Polyhedra
A Mystery of the Egyptian Calendar
A DodecahedronIcosahedron Doctrine
Johannes Kepler: from “Mysterium” to “Harmony”
A Regular Icosahedron as the Main Geometrical Object
of Mathematics
3.7. Regular Polyhedra in Nature and Science
3.8. Applications of Regular Polyhedrons in Art
3.9. Application of the Golden Mean in Contemporary Art
3.10. Conclusion
137
144
148
152
154
160
163
172
179
182
Part II. Mathematics of Harmony
Chapter 4. Generalizations of Fibonacci Numbers and the Golden Mean
4.1.
4.2.
4.3.
4.4.
A Combinatorial Approach to the Harmony of Mathematics
Binomial Coefficients and Pascal Triangle
The Generalized Fibonacci pNumbers
The Generalized Golden pSections
186
189
192
199
Contents
xv
4.5. The Generalized Principle of the Golden Section
4.6. A Generalization of Euclid’s Theorem II.11
4.7. The Roots of the Generalized Golden Algebraic Equations
4.8. The Generalized Golden Algebraic Equations of Higher Degrees
4.9. The Generalized Binet Formula for the Fibonacci рNumbers
4.10. The Generalized Lucas pNumbers
4.11. The “Metallic Means Family” by Vera W. de Spinadel
4.12. Gazale Formulas
4.13. Fibonacci and Lucas mNumbers
4.14. On the mExtension of the Fibonacci and Lucas pNumbers
4.15. Structural Harmony of Systems
4.16. Conclusion
202
204
206
212
214
221
227
232
237
241
249
253
Chapter 5. Hyperbolic Fibonacci and Lucas Functions
5.1.
5.2.
5.3.
The Simplest Elementary Functions
255
Hyperbolic Functions
259
Hyperbolic Fibonacci and Lucas Functions (StakhovTkachenko’s
Definition)
264
5.4. Integration and Differentiation of the Hyperbolic Fibonacci
and Lucas Functions and their Main Identities
268
5.5. Symmetric Hyperbolic Fibonacci and Lucas Functions
(StakhovRozin Definition)
277
5.6. Recursive Properties of the Symmetric Hyperbolic
Fibonacci and Lucas Functions
280
5.7. Hyperbolic Properties of the Symmetric Hyperbolic Fibonacci
and Lucas Functions and Formulas for Their Differentiation
and Integration
283
5.8. The Golden Shofar
286
5.9. A General Theory of the Hyperbolic Functions
291
5.10. A Puzzle of Phyllotaxis
299
5.11. A Geometric Theory of the Hyperbolic Functions
301
5.12. Bodnar’s Geometry
307
5.13. Conclusion
313
Chapter 6. Fibonacci and Golden Matrices
6.1.
6.2.
6.3.
Introduction into Matrix Theory
Fibonacci QMatrix
Generalized Fibonacci QpMatrices
317
322
326
Alexey Stakhov
MATHEMATICS OF HARMONY
xvi
6.4. Determinants of the QpMatrices and their Powers
6.5. The “Direct” and “Inverse” Fibonacci Matrices
6.6. Fibonacci GmMatrices
6.7. Fibonacci Qp,mMatrices
6.8. Determinants of the Qp,mMatrices and their Powers
6.9. The Golden QMatrices
6.10. The Golden GmMatrices
6.11. The Golden Genomatrices by Sergey Petoukhov
6.12. Conclusion
330
333
334
340
343
345
348
350
357
Part III. Application in Computer Science
Chapter 7. Algorithmic Measurement Theory
7.1.
7.2.
7.3.
7.4.
7.5.
7.6.
7.7.
The Role of Measurement in the History of Science
360
Mathematical Measurement Theory
364
Evolution of the Infinity Concept
370
A Constructive Approach to Measurement Theory
377
Mathematical Model of Measurement
382
Classical Measurement Algorithms
385
The Optimal Measurement Algorithms Originating Classical
Positional Number Systems
389
7.8. Optimal Measurement Algorithms Based on the Arithmetical
Square
392
7.9. Fibonacci Measurement Algorithms
396
7.10. The Main Result of Algorithmic Measurement Theory
401
7.11. Mathematical Theories Isomorphic to Algorithmic Measurement
Theory
408
7.12. Conclusion
413
Chapter 8. Fibonacci Computers
8.1.
8.2.
8.3.
8.4.
8.5.
8.6.
A History of Computers
Basic Stages in the History of Numeral Systems
Fibonacci pCodes
Minimal Form and Redundancy of the Fibonacci pCode
Fibonacci Arithmetic: The Classical Approach
Fibonacci Arithmetic: An Original Approach
416
424
429
434
443
449
Contents
xvii
8.7. Fibonacci Multiplication and Division
8.8. Hardware Realization of the Fibonacci Processor
8.9. Fibonacci Processor for Noisetolerant Computations
8.10. The Dramatic History of the Fibonacci Computer Project
8.11. Conclusion
455
460
465
470
475
Chapter 9. Codes of the Golden Proportion
9.1.
9.2.
Numeral Systems with Irrational Bases
476
Some Mathematical Properties of the Golden pProportion
Codes
480
9.3. Conversion of Numbers from Traditional Numeral Systems into
the Golden pProportion Codes
484
9.4. Golden Arithmetic
488
9.5. A New Approach to the Geometric Definition of a Number
492
9.6. New Mathematical Properties of Natural Numbers (Z and
Dproperties)
497
9.7. The F and LCodes
500
9.8. Numbertheoretical Properties of the Golden pProportion Codes 505
9.9. The Golden Resistor Dividers
511
9.10. Application of the Fibonacci and Golden Proportion Codes
to DigitaltoAnalog and AnalogtoDigital Conversion
515
9.11. Conclusion
520
Chapter 10. Ternary MirrorSymmetrical Arithmetic
10.1.
10.2.
10.3.
10.4.
10.5.
10.6.
A Ternary Computer “Setun”
TernarySymmetrical Numeral System
TernarySymmetrical Arithmetic
Ternary Logic
Ternary MirrorSymmetrical Representation
The Range of Number Representation and Redundancy
of the Ternary MirrorSymmetrical Numeral System
10.7. MirrorSymmetrical Summation and Subtraction
10.8. MirrorSymmetrical Multiplication and Division
10.9. Typical Devices of Ternary MirrorSymmetrical Processors
10.10. Matrix and Pipeline MirrorSymmetrical Summators
10.11. Ternary MirrorSymmetrical DigittoAnalog Converter
10.12. Conclusion
523
527
530
533
538
544
546
553
557
561
565
567
Alexey Stakhov
MATHEMATICS OF HARMONY
xviii
Chapter 11. A New Coding Theory Based on a Matrix Approach
11.1. A History of Coding Theory
11.2. Nonsingular Matrices
11.3. Fibonacci Encoding/Decoding Method Based upon Matrix
Multiplication
11.4. The Main Checking Relations of the Fibonacci
Encoding/Decoding Method
11.5. Error Detection and Correction
11.6. Redundancy, Correcting Ability, and the Advantages of the
Fibonacci Encoding/Decoding Method
11.7. Matrix Cryptography
11.8. Conclusion
569
579
581
584
588
597
601
613
Epilogue. Dirac’s Principle of Mathematical Beauty and the Mathematics
of Harmony: Clarifying the Origins and Development of Mathematics
E.1. Introduction
615
E.2. The “Strategic Mistakes” in the Development of Mathematics
620
E.3. Three “Key” Problems of Mathematics and a New Approach to
the Mathematics Origins
632
E.4. The Generalized Fibonacci Numbers and the Generalized Golden
Proportions
633
E.5. A New Geometric Definition of Number
641
E.6. Fibonacci and “Golden” Matrices
644
E.7. Applications in Computer Science: the “Golden” Information
Technology
646
E.8. Fundamental Discoveries of Modern Science Based Upon the
Golden Section and “Platonic Solids”
649
E.9. Conclusion
657
References
661
Appendix: Museum of Harmony and Golden Section
675
Index
685
Introduction
xix
Introduction
Algebra and Geometry have one and the
same fate. The very slow successes did
follow after the fast ones at the begin
ning. They left science in a state very far
from perfect. It happened, probably, be
cause mathematicians paid the main at
tention to the higher parts of the Analy
sis. They neglected the beginnings and
did not wish to develop those fields,
which they finished once and left them
from behind.
Nikolay Lobachevsky
Three “Key” Problems of Mathematics on the
Stage of its Origin
1. The Main Stages of Mathematics Development
W
hat is mathematics? What are its origin and history? What distin
guishes mathematics from other sciences? What is the subject of math
ematical research today? How does mathematics influence the development
of other sciences? To answer these questions we refer to the book Mathemat
ics in its Historical Development [1], written by the phenomenal Russian math
ematician and academician, Andrew Kolmogorov. According to Kolmogor
ov’s definition, mathematics is “a science about quantitative relations and spa
tial forms of real world.”
Kolmogorov writes that “the clear understanding of mathematics, as a spe
cial science having its own subject and method, arose for the first time in An
Alexey Stakhov
MATHEMATICS OF HARMONY
xx
cient Greece at 65 centuries BC after the accumulation of the big enough
actual material.”
Kolmogorov points out the following stages in mathematics de
velopment:
1. Period of the “Mathematics origin,” which preceded Greek mathe
matics.
2. Period of the “Elementary Mathematics.” This period started during
65 centuries BC and ended in the 17th century. The volume of
mathematical knowledge obtained up to the beginning of 17th cen
tury was, until now, the base of “elementary mathematics”—which
is taught at the secondary and high school levels.
3. The “Higher Mathematics” period, started with the use of variables
in Descartes’ analytical geometry and the creation of differential
and integral calculus.
4. The “Modern Mathematics” period. Lobachevsky’s “imaginary geom
etry” is considered the beginning of this period. Lobachevsky’s ge
ometry was the beginning of the expansion of the circle of quantita
tive relations and spatial forms—which began to be investigated by
mathematicians. The development of a similar kind of mathemati
cal research gave mathematicians many new important features.
2. A “Count Problem”
Discussing the reasons of mathematical occurrence, Kolmogorov specifies two
practical problems that stimulated the development of mathematics during
its origin: count and measurement.
A “count problem” was the first ancient problem of mathematics. It is em
phasized [1] that “on the earliest steps of culture development, the count of
things led to the creation of the elementary concepts of natural number arith
metic. On the base of the developed system of oral notation, written notations
arose, whereby different methods of the fulfillment of the four arithmetical
operations for natural numbers were gradually developed.”
The period that culminated in the origin of mathematics germinated the
“key” mathematical discoveries. We are talking about the positional princi
ple of numbers representation. It is emphasized in [2] that “the Babylonian
sexagesimal numeral system, which arose approximately in 2000 BC, was the
first numeral system based on the positional principle.” This discovery under
lies all early numeral systems created during the period of mathematics origin
Introduction
xxi
and the period of the elementary mathematics (including decimal and binary
systems).
It is necessary to note that the positional principle of number representa
tion and positional numeral systems (particularly the binary system), which
were created in the period of mathematics origin, became one of the “key”
ideas of modern computers. In this connection, it is also necessary to remem
ber that multiplication and division algorithms, used in modern computers,
were created by the ancient Egyptians (the method of doubling) [2].
However, the formation of the natural number’s concept was the main
result of arithmetic’s development in the period of mathematics origin. Natu
ral numbers are one of the major and fundamental mathematical concepts—
without which the existence of mathematics is impossible. For studying the
properties of natural numbers, the number theory—one of the fundamental
mathematical theories—arose in this ancient period.
3. A “Measurement Problem”
Kolmogorov emphasizes in [1], that “the needs of measurement (of quantity
of grain, length of road, etc.) had led to the occurrence of the names and des
ignations of the elementary fractions and to the development of the methods
of the fulfillment of arithmetic operations for fractions.... The measurement of
areas and volumes, the needs of the building engineering, and a little bit later
the needs of astronomy caused the development of geometry”.
Historically, the first “theory of measurement” arose in ancient Egypt. It
was the collection of rules, which the Egyptian land surveyors used. As the
ancient Greeks testify, geometry—as a “science of Earth measurement”—had
originated from these rules.
However, a discovery of the “incommensurable line segments” was
the “key” discovery in this area. This discovery had been made in the 5th
century BC in Pythagoras’ scientific school at the investigation of the ra
tio of the diagonal to the side of a square. Pythagoreans proved that this
ratio cannot be represented in the form of the ratio of two natural num
bers. Such line segments were named incommensurable, and the numbers,
which represented similar ratios, were named “irrationals.” A discovery of
the “incommensurable line segments” became a turning point in the devel
opment of mathematics. Owing to this discovery, the concept of irrational
numbers, the second fundamental concept (after natural numbers) came
into use in mathematics.
Alexey Stakhov
MATHEMATICS OF HARMONY
xxii
For overcoming the first crisis in the bases of mathematics, caused by the
discovery of “incommensurable line segments,” the Great mathematician Eu
doxus had developed a theory of magnitudes, which was transformed later
into mathematical measurement theory [3, 4], another fundamental theory
of mathematics. This theory underlies all “continuous mathematics” includ
ing differential and integral calculus.
Influence of the “measurement problem” on the development of mathe
matics is so great that the famous Bulgarian mathematician L. Iliev had de
clared that “during the first epoch of mathematics development, from antiq
uity to the discovery of differential and integral calculus, mathematics, inves
tigating first of all the measurement problems, had created Euclidean geome
try and number theory” [5].
Thus, the two “key” problems of ancient mathematics, the count problem
and the measurement problem, had led to the formation of the two fundamen
tal concepts of mathematics: natural numbers and irrational numbers—which,
together with number theory and measurement theory, became the basis of
“classical mathematics.”
4. Mathematics. The Loss of Certainty
The book, Mathematics: The Loss of Certainty [6], written by American math
ematician Morris Kline, had a huge influence upon the author and became a
source of reflections about the nature and role of mathematics in modern sci
ence; it is a pleasure for the author to retell briefly the basic ideas of Morris
Kline’s book.
Since the origin of mathematics as an independent branch of knowledge
(Greek mathematics), and during more than two millennia, mathematics was
engaged in a search for truth and had achieved outstanding successes. It seemed
that the vast amount of theorems about numbers and geometrical figures, which
was proved in mathematics, is an inexhaustible source of absolute knowledge
which never can change.
To obtain surprisingly powerful results, mathematicians had used a spe
cial deductive method which allowed them to get new mathematical results
(theorems) from a small number of axiomatic principles, named by axioms.
The nature of the deductive method guarantees a validity of the conclusion if
the initial axioms are true. Euclid’s Elements became the first great mathe
matical work in this area, which is a brilliant example of the effective applica
tion of the deductive method.
Introduction
xxiii
Euclidean geometry became the most esteemed part of mathematics—not
only because the deductive construction of mathematical disciplines had be
gun with the Euclidean geometry—but, also its theorems completely corre
sponded to the results of physical research. It was considered a firm scientific
axiom for many millennia. Euclidean geometry is the geometry of the physical
world surrounding us. That is why the unusual geometries created at the be
ginning of the 19th century, named nonEuclidean geometries, became the
first “blow” to the harmonious building of mathematical science. These un
usual geometries had forced mathematicians to recognize that mathematical
theories and theorems are not absolute truths in application to Nature. It was
proved that new geometries are mathematically correct, that is, they could be
geometrical models of the real world similar to Euclidean geometry, but then
the following question arises: what geometry is a true model of the real world?
Finding the contradictions in Cantor’s theory of infinite sets was another
“blow” to mathematics. Comprehension of the “Tsarina of sciences” is not perfect
regarding its structure; it lacks much, and it is subjected to monstrous contradic
tions, which can appear at any moment; it shocked mathematicians. The reaction
of mathematicians to all of these events was ambiguous. Unfortunately, the ma
jority of mathematicians had simply decided to ignore these contradictions. In
stead, they fenced themselves off from the external world and concentrated their
efforts on the problems arising within the modern field of mathematics, that is,
mathematicians decided to break connections with natural sciences.
What was mathematics during several millennia? For previous genera
tions, mathematics was first of all of the greatest creation of human intellect
intended for nature’s research. The natural sciences were the flesh and blood
of mathematics and it fed mathematicians with their vivifying juices. Mathe
maticians willingly cooperated with physicists, astronomers, chemists, and en
gineers in searching for the solution to various scientific and technical prob
lems. Moreover, many great mathematicians of the past were often outstand
ing physicists and astronomers. The mathematics was the “Tsarina” and si
multaneously the “Servant” of natural sciences.
Morris Kline noticed that “pure” mathematics, which had completely dis
associated from the inquiries of natural sciences, was never the center of at
tention and interest of the great mathematicians of the past. They considered
“pure” mathematics as some kind of “entertainment,” a rest from much more
important and fascinating problems, which were put forward by natural sci
ences. In the 18th century, such abstract science like number theory had in
volved only a few mathematicians. For example, Euler, whose scientific inter
ests had been connected with number theory, was the first to be a recognized
Alexey Stakhov
MATHEMATICS OF HARMONY
xxiv
specialist in mathematical physics. The Great mathematician Gauss did not
consider number theory as the major branch of mathematics. Many of his col
leagues suggested that he solves The Great Fermat Theorem. In one letter, Gauss
noted that Fermat’s hypothesis is an isolated mathematical problem, which is
not connected with the most important mathematical problems, and conse
quently, it is not of particular interest.
Morris Kline specifies the various reasons that induced mathematicians
to depart from studying the real world. Widening mathematical and natural
scientific research did not allow scientists to feel equally free in both mathe
matics and natural sciences. The problems, that stood before natural sciences
a solution to which the great mathematicians of the past participated active
ly nowadays became more and more complex, and many mathematicians had
decided to limit their activity to the problems of “pure” mathematics.
Abstraction, generalizations, specialization, and axiomatization are the
basic directions of activity chosen by “pure” mathematicians. This activi
ty led to the situation where, nowadays, mathematics and natural sciences
go different ways. New mathematical concepts are developing without any
attempt to find their applications. Moreover, mathematicians and repre
sentatives of natural sciences do not understand each other today—owing
to the excessive specialization in fields and often mathematicians do not
understand each other.
What can resolve this situation? Morris Kline emphasizes that researchers
should return to nature and natural sciences, which were the original objectives
of mathematics. Ultimately, common sense should win. The mathematical world
should search for a distinction not between “pure” and applied mathematics,
but between the mathematics; whereby, its purpose is to find a solution to rea
sonable problems. Mathematicians should not indulge someone’s personal tastes
and whims as our quests in mathematics is purposeful and neverending be
cause mathematics is rich in content that is empty, alive, and bloodless.
5. A “Harmony Problem”
As is known, returning to the past is a fruitful source of cognition to the present.
The return to the sources of mathematics, to its history, is one of the impor
tant directions to overcome the crisis of contemporary mathematics. In re
turning to ancient science, particularly Greek science, we should pay atten
tion to an important scientific problem, which was the focus of ancient sci
ence starting with Pythagoras and Plato.
Introduction
xxv
We are talking about the “harmony problem.” What is the harmony?
Wellknown Russian philosopher Shestakov in his remarkable book Har
mony as an Aesthetic Category [7] emphasizes that “in the history of aesthet
ic doctrines, the diversified types of understanding of harmony were put
forward. The concept of “harmony” is multiform and used extremely widely.
It meant the natural organization of nature and space, a beauty of the hu
man physical and moral world, principles of art works’ design or the law of
aesthetic perception.” Among the various types of harmony (mathematic,
aesthetic, artistic), which arose during the development of science and aes
thetics, we will first be interested in mathematical harmony. In this sense,
harmony is understood as equality or proportionality of the parts between
themselves and the parts with the whole. In the Great Soviet Encyclopedia,
we can find the following harmony definition, which expresses the mathe
matical understanding of the harmony: “The harmony of an object is a pro
portionality of the parts and the whole, a merge of the various components
of the object to create a uniform organic whole. In harmony, the internal
order and the measure of the object had obtained external revealing.”
In the present book we concentrate our attention on mathematical harmo
ny. It is clear that the mathematical understanding of harmony accepts, as a
rule, the mathematical kind, and it is expressed in the form of certain numer
ical proportions. Shestakov emphasizes [7] that mathematical harmony “fixes
attention on its quantitative side and is indifferent to qualitative originality
of the parts forming conformity... The mathematical understanding of the har
mony fixes, first of all, quantitative definiteness of the harmony, but it does
not express aesthetic quality of the harmony, its expressiveness, connection
with a beauty.”
6. The Numerical Harmony of the Pythagoreans
Pythagoreans, for the first time, put forth the idea of harmonious organiza
tion of the universe. According to Pythagoreans, “harmony is an internal con
nection of the things, without which the Cosmos could not exist.” At last,
according to Pythagoras, harmony has numerical representation, namely that
harmony is connected with the concept of number. The Pythagoreans had
created the doctrine about the creative essence of number and their number
theory had a qualitative character. Aristotle, in his “Metaphysics”, emphasiz
es this feature of the Pythagorean doctrine:
Alexey Stakhov
MATHEMATICS OF HARMONY
xxvi
“The socalled Pythagoreans, studying mathematical sciences, for the
first time have moved them forward and, basing on them, began to
consider mathematics as the beginnings of all things... Because all
things became like to numbers, and numbers occupied first place in
all nature, they assumed that the elements of numbers are the begin
ning of all things and that all universe is harmony and number.”
Pythagoreans recognized that a form of the universe should be harmoni
ous, and all elements of the universe are connected with harmonious figures.
Pythagoras taught that the Cube originates the Earth, the Tetrahedron the
Fire, the Octahedron the Air, the Icosahedron the Water, the Dodecahedron
the sphere of the universe, that is, the Ether.
The Pythagorean doctrine about the numerical harmony of the universe
had influenced the development of all subsequent doctrines about the nature
and essence of harmony. It was reflected upon and developed in the works of
great thinkers. In particular, the Pythagorean doctrine underlies Plato’s cos
mology. Plato developed the Pythagorean doctrine; specifically emphasizing the
cosmic importance of harmony. He remained firmly convinced that world har
mony can be expressed in numerical proportions. The influence of Pythagore
ans is especially traced in Plato‘s “Timaeus”; whereby, Plato developed the doc
trine about proportions and analyzed the role of Regular Polyhedrons (Platon
ic Solids), from which—in his opinion—God had created the world.
The main conclusion, which follows from the Pythagorean doctrine, con
sists of the following. Numerical or mathematical harmony is objective prop
erty of the universe, it exists irrespective of our consciousness and is expressed
in the harmonious organization of all in the real world starting from cosmos
and finishing by microcosm.
7. A “Harmony Problem” in Euclid’s Elements
We ask how Pythagoras and Plato’s harmonious ideas were reflected in antique
mathematics. To answer this question we analyze the greatest mathematical
work of Greek mathematics: the Elements of Euclid. As is known, the Elements
of Euclid is not an original work. A significant part of Elements was written by
Pythagorean mathematicians. Their contribution to the theory of proportions—
in which all ancient science and culture is based—is especially great. As the
further progression of science had shown, the Pythagoreans, using numerical
representations, did not leave the real world, but rather came nearer to it.
Introduction
xxvii
The 13th, and final, Book of Euclid’s Elements is devoted to the theory of
the regular polyhedrons, which is expressed in ancient science as universe har
mony. The regular polyhedrons were used by Plato in his Cosmology and there
fore they were named Platonic Solids. This fact originated the widespread hy
pothesis formulated by Proclus—one of the most known commentators of
Euclid’s Elements. According to Proclus’ opinion, Euclid created the Elements
not with the purpose to present geometry as axiomatic mathematical science,
but with the purpose to give the full systematized theory of Platonic Solids, in
passing having covered some advanced achievements of the ancient mathe
matics. Thus, the main goal of the Elements was a description of the theory of
Platonic Solids described in the final book of Elements. It would not be out of
place to remember that seemingly, the most important material of a scientific
book is placed into the final Chapter of the book. Consequently, the place
ment of the Platonic Solids theory in the final book of the Elements is indirect
proof surrounding the validity of Proclus’ hypothesis; meaning that Pythago
ras’ Doctrine about the numerical harmony of the universe got its brightest
embodiment in the greatest mathematical work of the ancient science: Eu
clid’s Elements.
In order to develop a complete theory of the Platonic Solids, in particular
the Dodecahedron, Euclid formulated in Book II the famous Theorem II,11
about the division in the extreme and mean ratio (DEMR), which is known in
modern science under the name of the golden section. DEMR penetrated all
Books of Euclid’s Elements, and it had been used by Euclid for the geometric
construction of the following “harmonic” geometric figures: equilateral trian
gle with the angles 72°, 72° and 36° (the “golden” equilateral triangle), regular
pentagon and then the Dodecahedron based on the golden section. Taking
into consideration Proclus’ hypothesis, and a role of the DEMR in Euclid’s
Elements, we can put forward the following unusual hypothesis: Euclid’s Ele
ments was the first attempt to create the “Mathematical Theory of Har
mony” which was the main idea of Greek science.
It is clear that the formulation of the division in the extreme and mean
ratio (the golden section) can be considered as the “key” mathematical dis
covery in the field of the “harmony problem.” The Great Russian philosopher
Alexey Losev wrote in one of his articles that: “From Plato’s point of view,
and generally from the point of view of all antique cosmology, the universe is
a certain proportional whole that is subordinated to the law of harmonious
division, the Golden Section.”
Thus, we have to add the “harmony problem” to the list of the “key” prob
lems of mathematics regarding the stage of its origin. Such approach leads us
Alexey Stakhov
MATHEMATICS OF HARMONY
xxviii
to the original view on the history of mathematics. This idea underlies the
present book.
During its historical development, the “classical mathematics” had lost
Pythagoras’ and Plato’s “harmonious idea” embodied by Euclid in his Ele
ments. As the outcome, mathematics had been divided into a number of math
ematical theories (geometry, number theory, algebra, differential and integral
calculus, etc.), which sometimes have very weak correlations. Unfortunately,
a significance of the “golden mean” had been belittled in modern mathematics
and theoretical physics. For many modern mathematicians, the “golden sec
tion” reminds us of a “beautiful fairy tale,” which has no relation to serious
mathematics.
8. Fibonacci Numbers
Nevertheless, despite the negative relation of “materialistic” mathematics
to the “golden mean,” its theory continued to develop. The famous Fibonac
ci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, …, had been introduced into mathe
matics during the 13th century by the famous Italian mathematician Le
onardo from Pisa (Fibonacci) at the solution of the rabbits reproduction
problem. It is necessary to note that the method of recursive relations—
one of the most powerful methods of combinatorial analysis—follows di
rectly from Fibonacci’s discovery. Later, the Fibonacci numbers had been
found in many natural objects and phenomena, in particular, the botanical
phenomenon of phyllotaxis.
9. The First Book on the Golden Mean in the History of Science
During the Italian Renaissance, interest in the “golden mean” arose with new
force. Of course, the universal genius of the Italian Renaissance Leonardo da
Vinci could not pass the division of the extreme and mean ratio (the golden
section). There is an opinion that Leonardo had introduced into the Renais
sance culture by the name of the “golden section.” Leonardo da Vinci had
influenced the book Divina Proportione [8], which was published by Italian
mathematician Luca Paccioli in 1509. This unique book was the first mathe
matical book on the “golden mean” in history. The book was illustrated with
60 brilliant geometric figures drawn by Leonardo da Vinci; additionally, the
book had a great influence on Renaissance culture.
Introduction
xxix
10. Johannes Kepler and the Golden Section
In the 17th century, astronomer and mathematician Johannes Kepler created
the original geometrical model of Solar system based on Platonic Solids. Ke
pler had expressed his admiration of the golden section with the following
words: “Geometry has two great treasures: one is the Theorem of Pythagoras;
the other, the division of a line into extreme and mean ratio. The first, we may
compare to a measure of gold; the second we may name a precious stone.”
11. Fibonacci Numbers and the Golden Section in 19th Century Science
After Kepler’s death, interest in the golden section, considered one of the two
“treasures of geometry,” decreased; whereby, such strange oblivion continued
for two centuries. Active interest in the golden section revived in mathemat
ics in the 19th century. During this period, many mathematical works were
devoted to Fibonacci numbers and the golden mean, and according to the
witty saying of one mathematician: they “started to reproduce as Fibonacci’s
rabbits.” French mathematicians Lucas and Binet became the leaders of this
type of research in 19th century. Lucas had introduced into mathematics the
name “Fibonacci Numbers,” and also the famous Lucas numbers (1, 3, 4, 7, 11,
18, ...). Binet had deduced the famous Binet formulas, which connect the Fi
bonacci and Lucas numbers with the golden mean.
During this time, the German mathematician Felix Klein tried to unite
together all branches of mathematics on the base of the Regular Icosahedron,
the Platonic Solid—dual to the Dodecahedron. Klein treats the Regular Icosa
hedron based on the golden section as the main geometric object, from which
the branches of the five mathematical theories follow, namely, geometry, Ga
lois’ theory, group theory, invariant theory, and differential equations. Klein’s
main idea is extremely simple: “Each unique geometrical object is somehow or
another connected to the properties of the Regular Icosahedron.”
12. The Golden Section and Fibonacci Numbers in Science of the 20th and
21st Centuries
In the second half of the 20th century the interest in Fibonacci numbers and the
golden mean in mathematics had revived with new force, and the revival expanded
Alexey Stakhov
MATHEMATICS OF HARMONY
xxx
into the 21st century with many original books [957] being published that were
devoted to the golden mean, Fibonacci numbers, and other related topics, which
is evidence of the increasing interest in the golden mean and Fibonacci numbers
in modern science. Prominent mathematicians Gardner [12], Vorobyov [13],
Coxeter [14], and Hoggatt [16] were the first researchers who felt new tenden
cies growing in mathematics. In 1963, the group of American mathematicians
had organized the Fibonacci Association and they started publishing the math
ematical journal The Fibonacci Quarterly. Owing to the activity of the Fibonacci
Association and the publications of the special books by Vorobyov [13], Hog
gatt [16], Vaida [28], Dunlap [38], and other mathematicians, a new mathemat
ical theory—the “Fibonacci numbers theory”—appeared in contemporary math
ematics. This theory has its own interesting mathematical history, which is pre
sented in the book A Mathematical History of the Golden Number, written by
the prominent Canadian mathematician Roger HerzFishler [40].
In 1992 a group of the Slavic scientists from Russia, Ukraine, Belarus, and
Poland had organized the socalled Slavic “Golden” Group. Resulting from
the initiative of this group, the International symposiums of “The Golden
Section and Problems of System Harmony” had been held in Kiev, Ukraine
in 1992 and 1993, and then again in Stavropol, Russia from 19941996.
The golden mean, pentagram, and Platonic Solids were widely used by
astrology and other esoteric sciences, and this became one of the reasons for
the negative reaction of “materialistic” science towards the golden mean and
Platonic Solids. However, all attempts of “materialistic” science and mathe
matics to forget and completely disregard the “golden mean” and Platonic
Solids and to throw them out along with astrology and esoteric sciences on
the “dump of the doubtful scientific concepts,” had failed. Mathematical models
based on the golden mean, Fibonacci numbers, and Platonic Solids had proved
to be “enduring,” and they began to appear unexpectedly in different areas of
nature. Already, Johannes Kepler had found Fibonacci’s spirals on the surface
of the phyllotaxis objects. The research of the phyllotaxis objects growth made
by the Ukrainian architect Oleg Bodnar [37, 52] demonstrated that the ge
ometry of phyllotaxis objects is based on a special class of hyperbolic func
tions—the “golden” hyperbolic functions. In 1984, the Byelorussian, philoso
pher Eduardo Soroko, had formulated the “Law of structural harmony of sys
tems” [25]. This law confirmed a general character of selforganized processes
in the system of any nature; it demonstrated that all selforganized systems
are based on the generalized golden pproportions. Shechtman’s quasicrys
tals, based on the Platonic icosahedron, and fullerenes (Nobel Prize of 1996),
were based on the Archimedean truncated icosahedron, had confirmed Felix
Introduction
xxxi
Klein’s great prediction about the fundamental role of the icosahedron in sci
ence and mathematics [58]. Ultimately, Petoukhov’s “golden” genomatrices
[59] did completed the list of modern outstanding discoveries based on the
golden mean, Fibonacci numbers, and the regular polyhedra.
It is possible to assume that the increasing interest in the golden mean
and Fibonacci numbers in modern theoretical physics and computer science
is one of the main features of 21st century science. Prominent theoretical phys
icist and engineering scientist Mohammed S. El Nashie is a world leader in
this field [6072]. El Nashie’s discovery of the golden mean in the famous phys
ical twoslit experiment—which underlies quantum physics—became a source
of many important discoveries in this area, in particular, the Einfinity theory.
In this respect, we mention the works of El Nashie’s numerous followers work
ing in theoretical physics [7383]. It is also necessary to note the contribution
of Slavic researchers to this important area. The book [53] written by the
Byelorussian physicist Vasyl Pertrunenko is devoted to the applications of
the golden mean in quantum physics and astronomy. In 2006, the book Meta
physics of the 21st century [57], edited by the famous Russian physicist and
theorist Y.S. Vladimirov was published. The book [57] consists of three chap
ters and the last chapter was devoted to the golden mean applications in mod
ern science. This chapter begins with two important articles [59, 84]. Sta
khov’s article [84] is devoted to the substantiation of “Harmony Mathemat
ics” as a new interdisciplinary direction of modern science. Petoukhov’s arti
cle [59] is devoted to the description of the important scientific discovery: the
“golden” genomatrices; which reaffirms the deep mathematical connection
between the golden mean and genetic code. The famous Russian physicist
Professor Vladimirov (Moscow University) finishes his book Metaphysics [85]
with the following words: “It is possible to assert that in the theory of elec
troweak interactions there are relations that coincide with the ‘Golden Sec
tion’ that play an important role in the various areas of science and art.”
In the second half of the 20th century multiple interesting mathematical
discoveries in the area of golden mean applications in computer science and
mathematics had been made [86119]. In 1956, the young American mathe
matician George Bergman made an important mathematical discovery in the
field of number systems [86]. We are talking about the number system with
irrational base (the golden mean) described in [86]. Modern mathematicians
had been so anxious of overcoming the crisis in the basis of mathematics that
they simply had not noticed Bergman’s discovery, which is, without doubt,
one of the greatest mathematical discoveries in the field of number systems
after the discovery by Babylonians of the positional principle of number repre
Alexey Stakhov
MATHEMATICS OF HARMONY
xxxii
sentation. Bergman’s number system was generalized by Alexey Stakhov who
developed in the book [24] more general class of the number systems with irra
tional radices named “Codes of the golden proportion.” Alexey Stakhov, in the
article [105], developed a new approach to geometric definition of real numbers
that is of great importance for number theory. In his article [87], and then in the
book [20], Stakhov developed the socalled Fibonacci codes. The codes of the
golden proportion and Fibonacci codes became a source of the Fibonacci com
puter project [30] developed in the Soviet Union. This computer project was an
original project, which was defended by 65 patents issued by the State Patent
ing Departments of the United States, Japan, England, Germany, France, Can
ada, and other countries [120131]. Parallel with Soviet computer science, work
continued on Fibonacci computers in the United States [132135]. In the works
[44, 103, 113, 114], a new class of square matrices, the generalized Fibonacci
matrices and the socalled “golden” matrices, was developed. This led to a new
kind of theory of coding and cryptography [44, 113, 114].
A new class of hyperbolic functions, the hyperbolic Fibonacci and Lucas
functions, introduced by Alexey Stakhov, Ivan Tkachenko, and Boris Rozin
[51, 98, 106, 116, 119], was another important modern mathematical discov
ery.
The beginning of the 21st century is characterized by a number of the
interesting events; all of which have a direct relation to Fibonacci numbers
and the golden mean. First of all, it is necessary to note that three Interna
tional Conferences on Fibonacci Numbers and their Applications were held
in the 21st century (Arizona, USA, 2002; Braunschweig, Germany, 2004; Cal
ifornia, USA, 2006). In 2003, the international conference Problems of Har
mony, Symmetry, and the Golden Section in Nature, Science and Art was
held in Vinnitsa, Ukraine following the initiative of the Slavic “Golden” Group,
which had transformed into the International Club of the Golden Section.
In 2005, the Academy of Trinitarizm (Russia) and the International Club of
the Golden Section, had organized the Institute of the Golden Section.
Intersecting the 20th and 21st centuries, Western and Slavic scientists
had published a number of scientific books in the field of the golden mean and
its applications. The most interesting of them are the following:
Dunlap R.A. The Golden Ratio and Fibonacci Numbers (1997) [38].
HerzFishler Roger. A Mathematical History of the Golden Number
(1998) [40].
Vera W. de Spinadel. From the Golden Mean to Chaos (1998) [42].
Introduction
xxxiii
Gazale Midhat J. Gnomon. From Pharaohs to Fractals (1999) [45].
Kappraff Jay. Connections. The Geometric Bridge between Art and Sci
ence (2001) [47].
Kappraff Jay. Beyond Measure. A Guided Tour Through Nature, Myth,
and Number (2002) [50].
Shevelev J.S. Metalanguage of the Living Nature (2000) (Russian)[46].
Petrunenko V.V. The Golden Section in Quantum States and its Astro
nomical and Physical Manifestations (2005) (Russian) [53].
Bodnar O.J. The Golden Section and NonEuclidean Geometry in Sci
ence and Art (2005) (Ukrainian) [52].
Soroko E. M. The Golden Section, Processes of Selforganization and
Evolution of System. Introduction into General Theory of System Har
mony (2006) (Russian) [56].
Stakhov A.P., Sluchenkova A.A.. Scherbakov I.G. The da Vinci Code
and Fibonacci Series (2006) (Russian) [55].
Olsen Scott. The Golden Section: Nature’s Greatest Secret (2006) [54].
This list confirms a great interest in the golden mean in 21st century science.
13. The Lecture: “The Golden Section and Modern
Harmony Mathematics”
By the end of the 20th century, the development of the “Fibonacci numbers
theory” was widening intensively. Many new generalizations of Fibonacci
numbers and the golden section had been developed [20]. Different unexpected
applications of Fibonacci numbers and the golden section particularly in the
oretical physics (the hyperbolic Fibonacci and Lucas functions [51, 98, 106]),
computer science (Fibonacci codes and the codes of the golden proportion
[20, 24, 87, 89, 102]), botany (the law of the spiral biosymmetries transforma
tion [37]), and even philosophy (the law of structural harmony of systems
[25]) were obtained. It became clear that the new results in this area were far
beyond the traditional “Fibonacci numbers theory” [13, 16, 28]. Moreover, it
became evident that the name “Fibonacci numbers theory” considerably nar
rows the subject of this scientific direction—which studies mathematical mod
els of system harmony. Therefore, the idea to unite the new results in the theory
Alexey Stakhov
MATHEMATICS OF HARMONY
xxxiv
of the golden mean and Fibonacci numbers and their applications under the flag
of the new interdisciplinary direction of the modern science, named “Harmony
Mathematics,” appeared. Such idea had been presented by Alexey Stakhov in
the lecture “The Golden Section and Modern Harmony Mathematics” at the
Seventh International Conference on Fibonacci Numbers and their Applications
in Graz, Austria in July 1996. The lecture was later published in the book Appli
cations of Fibonacci Numbers [100].
After 1996, the author continued to develop and deepen this idea [101
119]. However, the creation of “Harmony Mathematics” was a result of col
lective creativity; the works of other prominent researchers in the field of the
golden section and Fibonacci numbers Martin Gardner [12], Nikolay Voro
byov [13], H. S. M. Coxeter [14], Verner Hoggat [16], George Polya [17],
Alfred Renyi [23], Stephen Vaida [28], Eduardo Soroko [25, 56], Jan Grzedz
ielski [26], Oleg Bodnar [37, 52], Nikolay Vasutinsky [31], Victor Korobko
[43], Josef Shevelev [46], Sergey Petoukhov [59], Roger HerzFishler [40],
Jay Kappraff [47, 50], Midhat Gazale [45], Vera W. de Spinadel [42], R.A.
Dunlap [38], Scott Olsen [54], Mohammed S. El Nashie [6072], and other
scientists had influenced the author’s research in this field.
“Harmony Mathematics,” in its origin, goes back to the Euclidean problem
of “division in the extreme and mean ratio” (the golden section) [40]. Harmony
Mathematics is a continuation of the traditional “Fibonacci numbers theory”
[13, 16, 28]. What are the purposes of this new mathematical theory? Similar to
“classical mathematics,” which is defined sometimes as the “science about mod
els” [5], we can consider Harmony Mathematics as the “science about the
models of harmonic processes” in the world surrounding us.
14. Two Historical Ways of Mathematics Development
In research, returning to the origin of mathematics, we can point out the two
ways of mathematics development, which arose in the ancient mathematics.
The first way was based on the “count problem” and the “measurement prob
lem” [1]. In the period of mathematics origin two fundamental discoveries
were made. The positional principle of number representation [2] was used
in all known numeral systems, including the Babylonian sexagecimal, deci
mal, and binary systems. Ultimately, the development of this direction culmi
nated in the formation of the concept of natural numbers; it also led to the
creation of number theory—the first fundamental theory of mathematics. In
commensurable line segments discovered by Pythagoreans led to the dis
Introduction
xxxv
covery of irrational numbers and the creation of measurement theory
[3, 4]—which was the second fundamental theory of mathematics. Ultimate
ly, natural and irrational numbers became those basic mathematical concepts
that underlie all mathematical theories of “classical mathematics,” including num
ber theory, algebra, geometry, and differential and integral calculus. Theoretical
physics and computer science are the most important applications of “classical
mathematics” (see Fig. I.1).
The “key” problems of the ancient mathematics
Count
problem
Measurement
problem
Harmony
problem
Positional principle of
number
representation
Incommensurable
line segments
Division in the
extreme and mean
ratio
Number theory
and natural
numbers
Measurement theory
and irrational
numbers
Theory of
Fibonacci
numbers and the
golden section
Classical mathematics
Theoretical physics
Computer science
Harmony Mathematics
The “golden” theoretical
physics
The “golden” computer
science
Figure I.1. Three “key” problems of the ancient mathematics and new directions in
mathematics, theoretical physics and computer science
However, parallel with the “classical mathematics” in the ancient science
Alexey Stakhov
MATHEMATICS OF HARMONY
xxxvi
another mathematical theory—Harmony Mathematics—had started to devel
op. Harmony Mathematics originated from another “key” idea of antique sci
ence—the “Harmony problem.” The Harmony Problem triggered the “Doc
trine about Numerical Harmony of the Universe” that was developed by
Pythagoras.
A division in the extreme and mean ratio (the golden section) was the
“key” mathematical discovery in this area [40]. The development of this idea
resulted in the Fibonacci numbers theory [13, 16, 28] in modern mathemat
ics. However, the extension of the Fibonacci numbers theory and its applica
tions coupled with the generalization of Fibonacci numbers and the golden
section produced the concept of “Harmony Mathematics” [100] as a new
interdisciplinary direction of modern science and mathematics. This can re
sult in the creation of the “golden” theoretical physics, based on the “gold
en” hyperbolic models of nature [51, 98, 106, 116, 118], and the “golden”
computer science, based on new computer arithmetic [20, 24, 30, 87, 89, 94,
104] and a new theory of coding and cryptography [55, 113, 114].
15. The Main Goal of the Present Book
It seems that the dramatic history of the DEMR (the golden section) that
continued over several millennia has ended as a great triumph for the golden
section in the beginning of the 21st century. Many outstanding scientific dis
coveries that are based on the golden section (quasicrystals, fullerenes , “gold
en” genomatrices and so on) gave reason to conclude that the golden section
may be considered as some kind of “metaphysical knowledge,” “prenum
ber,” or “universal code of Nature,” which could become the basis for the
future development of science; particularly, theoretical physics, genetics,
and computer science. This idea is the main concept of the book [57] and the
articles [59, 84]. These scientific facts demand reappraisal of the role of the
golden section in contemporary mathematics.
The main purpose of the present book is to revive the interest in the gold
en section and Pythagoras, Plato, and Euclid’s “harmonic idea” in modern
mathematics, theoretical physics, and computer science. It also strives to dem
onstrate that the Euclidean problem of the “division in extreme and mean
ratio” (the golden section) is a powerful and fruitful source of many funda
mental ideas and concepts of contemporary mathematics, theoretical physics,
and computer science. We consider different generalizations of the golden
mean, in particular, the generalized golden pproportions (p=0, 1, 2, 3, …) and
Introduction
xxxvii
the generalized golden means of the order m (m is a positive real number) as
fundamental mathematical constants similar to the numbers π and e. We show
that this approach resulted in: a new class of elementary functions—the hy
perbolic Fibonacci and Lucas functions; a new class of the recursive numerical
sequences—the generalized Fibonacci and Lucas pnumbers (p=0, 1, 2, 3, …)
and the generalized Fibonacci and Lucas numbers of the order m (m is a positive
real number); and it also led to a new class of square matrices—the Fibonacci
and “golden” matrices. Also, this approach resulted in a new measurement the
ory, algorithmic measurement theory, in a new class of number systems with
irrational radices that are codes of the golden pproportions. Additionally, a
new kind of computer arithmetic, the Fibonacci and “golden” arithmetic and
the ternary mirrorsymmetric arithmetic, was developed, as well as a new cod
ing theory based on the Fibonacci matrices and a new kind of cryptography—
the “golden” cryptography.
The book consists of three parts. Part I “Classical Golden Mean, Fibonacci
numbers, and Platonic Solids” consists of three chapters, Chapter 1 “The Gold
en Section”, Chapter 2 “Fibonacci and Lucas Numbers”, and Chapter 3 “Regu
lar Polyhedrons”. Part I is popular introduction into the Fibonacci numbers
theory and its applications. Part I is intended for a wide audience including
mathematics teachers of secondary schools, students of colleges and universi
ties. Also, Part I can attract attention to the representatives of various branches
of modern science and art that are interested in both creative and practical ap
plications of the golden mean, Fibonacci numbers, and Platonic Solids.
Part II “Mathematics of Harmony” consists of three chapters, Chapter 4
“Generalizations of Fibonacci Numbers and the Golden Mean,” Chapter 5
“Hyperbolic Fibonacci and Lucas Functions,” and Chapter 6 “Fibonacci and
Golden Matrices”. Part II calls for special knowledge in mathematics and is
intended, first of all, for mathematicians and scientists in theoretical physics.
Part III “Applications in Computer Science” consists of five chapters,
Chapter 7 “Algorithmic Measurement Theory”, Chapter 8 “Fibonacci Com
puters”, Chapter 9 “Codes of the Golden Proportion” , Chapter 10 “Ternary
MirrorSymmetrical Arithmetic,” and Chapter 11 “A New Coding Theory
Based on Matrix Approach.” Part III is intended for mathematicians and spe
cialists in computer science.
Note that Parts II and III are, in the main, results of original researches
obtained by the author in about 40 years of scientific work.
This page intentionally left blank
Acknowledgements
xxxix
Acknowledgements
This book is a result of my lifetime research in the field of the Golden Section,
Fibonacci numbers, “Mathematics of Harmony,” and their applications in mod
ern science. I have met along the way a lot of extraordinary people who helped
me to broaden and deepen my scientific knowledge and professional skills. More
than 40 years ago I read the remarkable brochure Fibonacci Numbers written by
the Russian mathematician Nikolay Vorobyov. This brochure was the first
mathematical work on Fibonacci numbers published in the second half of the
20th century. It determined my scientific interests in Fibonacci numbers. In
1974, I met with Professor Vorobyov in St. Petersburg (formerly Leningrad)
and narrated him about my scientific achievements in this area. He gave me as a
keepsake his brochure Fibonacci Numbers with the following inscription: “To
highly respected Alexey Stakhov with Fibonacci’s greetings.”
I have great gratitude to my teacher, the outstanding Ukrainian scientist,
Professor Alexander Volkov (19242007), under whose leadership I defend
ed my PhD dissertation in 1966 and DrSci dissertation in 1972. These disser
tations were the first step in my research, which led me to the new scientific
direction Mathematics of Harmony.
I appreciated a true value of mathematics as a “method of thinking” after
an influential collaboration with Igor Vitenko, a graduate of the mathemati
cal faculty of the Lvov University, who, without any doubts, would have been
known as an outstanding mathematician and glory of Ukrainian science if he
hadn’t passed away so early in life. His death in October 1974—which hap
pened as a result of suicide—was a big loss. Also, I consider the outstanding
Russian philosopher George Chefranov as my teacher of philosophy. Our
evening walks, along Chekhov’s street of Taganrog city, were accompanied
by hot discussions of philosophical problems of science, which became a major
philosophical university for me.
In my stormy scientific life I met many fine people, who could understand
me, my enthusiasm, and appreciate my scientific direction. With deep grati
tude, I recall a meeting in the Austrian city of Graz in 1976 with the excep
tional Austrian mathematician Alexander Aigner. His review of my lecture
at Graz University was the beginning of international recognition of my sci
entific direction. The German scientist and academician Dr. Volker Kempe
who now lives in Austria, also impacted on my career. Thanks to Dr. Kempe,
I was honored to be invited to Dresden Technical University as a visiting
Alexey Stakhov
MATHEMATICS OF HARMONY
xl
professor in 1988. Another remarkable person who had a great influence on
my work was the Ukrainian mathematician and academician Yury Mitropol
sky (19172008). His influence on my research pertinent to the history of
mathematics and other topics, such as the application of “Harmony Mathe
matics” to the contemporary mathematics and to the mathematical educa
tion, is inestimable contribution. Thanks to the support of Yury Mitropolski,
I was able to publish many scientific articles in various Ukrainian academic
journals.
My arrival in Canada in 2004 set the arena for the next stages of develop
ment in my scientific researches. Thanks to support from the famous physicist
Professor Mohammed El Nashie, the editorinchief of the international mag
azine Chaos, Solitons and Fractals, I was able to publish many articles in this
remarkable interdisciplinary journal. Within two years, I published 14 funda
mental articles in Chaos, Solitons and Fractals, which closed my cycle of re
search in the field of “Mathematics of Harmony.” These articles attracted at
tention of the international scientific community to my latest scientific re
sults; also, these articles paved the way for making new professional contacts
with worldrenowned Western scientists. Scientific discussion with outstand
ing scientists in the field of the Golden Section, such as the Argentinean math
ematician Vera W. de Spinadel—author of the book From the Golden Mean to
Chaos (1998), the American mathematician Louis Kauffman—editor of the
Knots and Everything Series from World Scientific, the American researcher
Jay Kappraff—author of two remarkable books Connections: The geometric
bridge between Art and Science (2001) and Beyond Measure: A Guided Tour
Through Nature, Myth, and Number (2002), have considerably influenced on
my research in the field of the Golden Section and Mathematics of Harmony.
Thanks to the support from Professors Louis Kauffman and Jay Kappraff,
this book was accepted by World Scientific for publication. I would like to
express my gratitude to the initiative of Professor Vera W. de Spinadel, who
helped with acceptance of my lecture, the “Three ‘Key’ Problems of Mathe
matics on the Stage of its Origin and the “Harmony Mathematics” as Alterna
tive Way of Mathematics Development,” as the plenary lecture at the Fifth
Mathematics & Design International Conference (July 2007, Brazil).
Furthermore, I would like to express special thanks to Dr. Scott Olsen,
the American philosopher and prominent researcher in the golden mean and
author of the book The Golden Section: Nature’s Greatest Secret (2006), for his
invaluable assistance in the scientific and English editing of my book. Also, I
would like to express my greatest gratitude to the Russian businessman and
scientist Dr. Ivan Raylyan, a great enthusiast of the Golden Section, for his
Acknowledgements
xli
financial assistance, which provided me with all of the best technology, infor
mational tools, and pecuniary means to produce a cameraready manuscript.
During my Canadian period of my scientific life I published a number of
important articles in the field of the Harmony Mathematics together with
Boris Rozin. I would like to express to him great gratitude for scientific col
laboration.
I would like to express my special thanks to Alexander Soypher for his
invaluable assistance in overall design of the book and illustrations, cover de
sign and color appendix “Museum of Harmony and Golden Section.“
Lastly, this book would never have been written without selfdenying sup
port of my wife Antonina, who always creates perfect conditions for my sci
entific work in any countries I have been; and who has been sailing with me
for more than 47 years on my “Golden” journey. In addition, I would like to
express my special appreciation to my daughter Anna Sluchenkova for her
critical remarks, and her invaluable help in the English translation and edit
ing of the book, and, especially, for her work in preparing illustrations, and
coordination and final preparation of cameraready manuscript. Without her
support this book would never been published.
Alexey Stakhov
This page intentionally left blank
Credits and Sources
xliii
Credits and Sources
Cover
Cover design credit: Alexander Soypher
Jacopo de Barbari, Portrait of Luca Pacioli source: Wikimedia
Commons, Galaxy M74 see credit for Figure 2.6b,
Nautilus spiral see credit for Figure 2.6a
Ch.1. p.15
Niels Abel portrait and A monument of Niels Abel in Oslo
source: www.abelprisen.no
Figure 1.6
Credit card image source: www.automotoportal.com
Figure 1.16 The Pentagon source: www.directionsmag.com
Figure 1.19 Pentagonal symmetry in nature credit Alexander Soypher
Figure 1.20 Hesire Panel from Shevelev I., Marutaev M., Shmelev I.,
Golden Section, 1990
Figure 1.21 Cheops Pyramid in Giza source:
www.bibleplaces.com/giza.htm
Figure 1.23 Doryphoros by Polycletes source:
www.goldensection.eu/abb10.jpg
Figure 1.24 Venus de Milo source: www.theoi.com
Figure 1.25 Holy Family by Michelangelo source:
britton.disted.camosun.bc.ca/goldslide/gold22.jpg
Figure 1.26 Crucifixion by Rafael Santi source:
britton.disted.camosun.bc.ca/goldslide/gold24.jpg
Ch.1. p.44
Leonardo da Vinci from Swedish journal
Svenska FamiljJournalen (18641887).
Ch.1. p.51
Memorial stone in Borgo San Sepolcro and In honor of 500
anniversary of Summa composite photo credit Alexander
Soypher; pictures sources:
utenti.quipo.it/base5/pacioli/statuapacioli.htm,
jeff560.tripod.com/stamps.html,
numismaticaitaliana.lamoneta.it/moneta/WITL/35
Figure 1.30 Saint Pietro in Montorio source:
www.wga.hu/art/b/bramante/ztempiet.jpg
Alexey Stakhov
MATHEMATICS OF HARMONY
xliv
Figure 1.31 Smolny Cathedral St.Petersburg, Russia source:
www.aditours.com/russiafinland.htm
Figure 1.33 Le Corbusier’s Modulor source:
britton.disted.camosun.bc.ca/goldslide/gold39.jpg
Figure 1.34 The Building of the United Nations Headquarters in New York
source:
www.inetours.com/New_York/Images/UN/UN_8874.jpg
Figure 1.35 Near the window by Konstantin Vasiliev source:
varvar.ru/arhiv/gallery/russian/vasilyev
Figure 1.36 The golden mean in human body (a) from Kovalev F., Golden
Section in Painting
The golden mean in clothes design (b) source:
www.olish.ru/picture/gold7.jpg
Figure 1.37 Harmonic analysis of woman face source:
www.beautyanalysis.com
Figure 1.38 Harmonic analysis of Nefertiti’s portrait from Soroko E.,
Golden code in sculpture portrait of Nefertiti
www.trinitas.ru/rus/doc/0232/004a/02320038.htm
Figure 2.1
Fibonacci’s rabbits source:
www.geider.net/esp/varios/numero_de_oro.htm
Figure 2.2
Family tree of honeybees source:
www.obscurious.co.uk/html/html.pages/Fibonacci.html
Figure 2.6a Nautilus spiral this Wikipedia and Wikimedia Commons
image is from the user Chris 73 and is freely available at
commons.wikimedia.org/wiki/File:
NautilusCutawayLogarithmicSpiral.jpg under the creative
commons ccbysa 2.5 license.
Figure 2.6b Galaxy M74 photo credit: Gemini Observatory/AURA,
source: www.gemini.edu
Figure 2.6c Sea Shell source: www.femorale.com.br
Figure 2.8
Crystal’s spatial lattices of NaCl (а) and CaO (b) source:
www.3dchem.com
Figure 2.11 Helical symmetry source: Dr. Ron Knott’s web page
www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html
Figure 2.13 Phyllotaxis structures:
Credits and Sources
xlv
(a) Cactus source: flickr.com/photos/wurzle/3668992
(b) Sunflower photograph by Janet Lanier from
www.athousandandone.com/photos/2/44b99d279626a_s.jpg
(c) Coneflower source:
www.wunderground.com/data/wximagenew/m/mentalno
mad/26.jpg
(d) Romanescu source: morgana.djo.tudelft.nl/romanescu.jpg
(e) Pineapple permission to use is granted by the GNU Free
Documentation License
(f) Pine cone source: flickr.com/photos/mrpattersonsir/11199034
Figure 2.14 Geometric models of phyllotaxis structures source: Jill Britton’s
web page britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm
Ch.2.
pp.130132
Edouard Zeckendorf, Verner Emil Hoggatt, Alfred Brousseau,
Herta Taussig Freitag source:
www.mscs.dal.ca/Fibonacci/biographies.html
Figure 3.4
Platonic Solids source: Wikimedia Commons, permission to
use is granted by the GNU Free Documentation License
Figure 3.5
Archimedean solids source: Wikimedia Commons, permission
to use is granted by the GNU Free Documentation License
Figure 3.6
A construction of the Archimedean truncated icosahedron from
E.Katz, Art and science – about polyhedrons in general and
about truncated icosahedron in particular
Figure 3.7
Cuboctahedron and icosidodecahedron source: Wikimedia
Commons, permission to use is granted by the GNU Free
Documentation License
Figure 3.8
Rhombicuboctahedron and rhombicosidodecahedron source:
Wikimedia Commons, permission to use is granted by the
GNU Free Documentation License
Figure 3.9
Snub hexahedron and snub dodecahedron source: Wikimedia
Commons, permission to use is granted by the GNU Free
Documentation License
Figure 3.10 KeplerPoinsot solids source: Wikimedia Commons, this image
is licensed under the Creative Commons Attribution
ShareAlike 2.5 License.
Figure 3.15 Egg of Life source: www.floweroflife.org/toverview6.htm
Ch.3. p. 169 Dan Shechtman source: pard.technion.ac.il
Alexey Stakhov
MATHEMATICS OF HARMONY
xlvi
Figure 3.21 Buckminster Fuller source: users.midwestmail.com/nightlife/
cdalerocks/nightlife/zzzPermanent/030821BuckyFuller.html
Fuller’s geodesic dome source:
www.jcsolarhomes.com/image/dome1.gif
Figure 3.22 The Montreal Biosphere source:
www.ec.gc.ca/EnviroZine/images/Issue59/Biosphere_l.jpg
Figure 3.23 The U.S. postage stamp source: www.bfi.org
Figure 3.25 Molecule C60 source:
www.3dchem.com/imagesofmolecules/c60.jpg
Figure 3.31 Piero della Francesco’s “The Baptism of Christ” the compilation
copyright is held by The Yorck Project and licensed under the
GNU Free Documentation License
Figure 3.32 Fra Giovanni da Verona’s intarsia source:
www.flickr.com/photos/powersmitchell/2249856120
www.flickr.com/photos/powersmitchell/2249856176
Figure 3.34 Abstract art of Astrid Fitzgerald permission to use is granted by
Astrid Fitzgerald, source: www.astridfitzgerald.com
Figure 3.35 The art works of Teja Krasek permission to use is granted by
Teja Krasek, source: tejakrasek.tripod.com
Figure 3.36 The Geometric Art of John Michell source:
www.thehope.org/jm_art.htm (Every attempt was made to
contact the author for permission, Mr. Michell died on April
24, 2009 in Poole, England)
Figure 3.37 Quantum Connections by Marion Drennen permission to use is
granted by Marion Drennen, source:
quantumconnectionsart.blogspot.com
Ch.6. p.318
James Joseph Sylvester source:
wwwhistory.mcs.stand.ac.uk/history/PictDisplay/Sylvester.html
Ch.8. p.422
John von Neumann credit United States Department of Ener
gy, source: Wikimedia Commons
Ch.11. p.570 Claude Elwood Shannon credit Bell Labs, source:
www.belllabs.com/news/2001/february/26/1.html
Ch.11. p.573 Richard Wesley Hamming credit Samuel P. Morgan, source:
cm.belllabs.com/cm/cs/alumni/hamming/index.html
Credits and Sources
xlvii
Appendix
Museum of Harmony and Golden Section
Symmetry in Nature
Butterfly credit Stavros Markopoulos, source:
www.flickr.com/photos/markop/273706600
Fern leaf credit Ken Ilio, source: www.flickr.com/photos/
kenilio/217816458
Tiger credit Brian Mckay, source: www.flickr.com/photos/
30775272@N05/2884963755
Peacock credit Gary Burke, source: www.flickr.com/photos/
gburke6/1806285477
Beetle credit Ivan Zheleznov, source: www.naturelight.ru/
show_photo/author429/9903.html
Cow credit Steve Arens, source: www.flickr.com/photos/
stevenarens/410707104
Black Swan credit Francesca Birini, source:
www.flickr.com/photos/firenzesca/1429628428
Kitten credit Erkhemchukhal Dorj, source: www.flickr.com/
photos/mongol/311285917
Cobra Snake credit Michael Allen Smith, source:
www.flickr.com/photos/digitalcolony/2289243736
Pentagonal symmetry in nature
Sea Urchin (on the background) underwater photo credit
Cornelia Kraus
Urchin shell source: www.fontplay.com/freephotos/imagesn/
fpfreefoto1519.jpg
Passion flower source: www.passionflowers.co.uk
Comb Sand Star credit Peter Bryant, source:
nathistoc.bio.uci.edu/Echinos/Astropecten.htm
Stapelia source: www.flickr.com/photos/21288646@N00/
285588504
Sea Star credit Paul Shaffner, source: www.flickr.com/photos/
paulshaffner/2232846436
Red Frangipani source: www.fontplay.com/freephotos/
4thfoldern/fp11040520.jpg
Alexey Stakhov
MATHEMATICS OF HARMONY
xlviii
Crown flower credit Jessy C.E, source: www.flickr.com/
photos/tropicaliving/1515336212
Periwinkle (Vinca minor) source: www.fontplay.com/
freephotos/nine/fpx04210712.jpg
White Frangipani source: www.fontplay.com/freephotos/
4thfoldern/fp11050518.jpg
Hoya Carnosa source: http://www.flickr.com/photos/flatpix/
1258040465
Golden spirals in nature
Barred Spiral Galaxy NGC 1300 credit NASA, ESA, and The
Hubble Heritage Team (STScI/AURA), source:
hubblesite.org
Sea shell source: www.flickr.com/photos/fdecomite/
2985635111
Fiddlehead fern credit James W. Gentry Jr., source:
www.flickr.com/photos/saynine/2348635947
Fern spiral (red) credit Faith Wilso, source: www.flickr.com/
photos/terrabella/2713519265
Nautilus shell see credit for Figure 2.6a
Aloe spiral credit J. Brew, source: www.flickr.com/photos/
brewbooks/184343329
Fiddleheads (green) credit Dave & Lesley, source:
www.flickr.com/photos/fireflies604/2834432521
Cala lily credit Dawn Endico, source: www.flickr.com/
photos/candiedwomanire/2886355
Snail source: www.flickr.com/photos/sailorganymede/
2884930696
Rose credit Phil Price, source: www.flickr.com/photos/
philprice/2610013436
Whirlpool Galaxy M51 credit NASA, ESA, S. Beckwith (STS
cI), and The Hubble Heritage Team (STScI/AURA), source:
hubblesite.org
The Spiralof Life credit Mr. Theklan, source: www.flickr.com/
photos/theklan/1161283822
Credits and Sources
xlix
Fibonacci numbers and phyllotaxis
Data palm crown credit Steve Flippen, source: ww.flickr.com/
photos/sdflip/307295569
Aloe flower credit Dan Martin, source: www.flickr.com/
photos/acurazinedan/386035959
Monstera Deliciosa flower credit Luidgi, source:
www.flickr.com/photos/luigistrano/2063302304
Pineapple source: puregreeni.us/wpcontent/uploads/2008/
02/450pxpineapple1.jpg
Banksia coccinea source: blog.paran.com/swan555/21036540
Norfolk Island Pine credit Brenda, source: www.flickr.com/
photos/omnia/67857983
Salak source: commons.wikimedia.org/wiki/File:Salak_01.jpg
Cycad cone credit Glenn Fleishman, source: www.flickr.com/
photos/glennf/20342176
Golden Section in architecture
United Nations Building source: www.inetours.com/
New_York/Pages/photos/UNHeadquartersbuilding.html
The Pentagon, US Department of Defense building credit US
Department of Defense, source: www.defenselink.mil/photos/
newsphoto.aspx?newsphotoid=1309
Smolny Cathedral in St. Petersburg source: www.lode.by/i/
photo/the_smolny_cathedral_174864.jpg
St. Andrew Cathedral in Kiev source: www.tour2kiev.com/
pub_img/0704161414_st_andrew.jpg
San Pietro in Montorio source: Wikimedia Commons
en.wikipedia.org/wiki/San_Pietro_in_Montorio
Notre Dame in Paris credit Andrea Schaffer, source:
www.flickr.com/photos/aschaf/2836231278
Cathedral of Our Lady of Chartres credit Dimitry B, source:
www.flickr.com/photos/ru_boff/520034494
Parthenon source: leftistmoon.files.wordpress.com/2007/10/
parthenon.jpg
Great pyramids of Giza credit Ricardo Liberato, this image is
Alexey Stakhov
MATHEMATICS OF HARMONY
l
licensed under Creative Commons Attribution ShareAlike 2.0
License, source: en.wikipedia.org/wiki/File:
All_Gizah_Pyramids.jpg
The Great Pyramid of Khufu source: whgbetc.com/mind/
pyramids_giza.jpg
Golden Section in Contemporary Abstract and Applied Art
Polyhedral Universe credit Teja Krasek
No. 258 from Cosmic Measures collection credit Astrid
Fitzgerald
Nine credit Marion Drennen
Double Pentagram Pentagons credit John Michell
Cosmic Gems credit Teja Krasek
Pentagonal Hexagons credit John Michell
No. 134 from Constructions collection credit Astrid Fitzgerald
Life, the Universe, and Everything credit Teja Krasek
No. 283 from All is Number: Number is All collection credit
Astrid Fitzgerald
One credit Marion Drennen
Four credit Marion Drennen
No.129 from Constructions collection credit Astrid Fitzgerald
Homage to Pythagoras credit Marion Drennen
This page intentionally left blank
Part I
Part I.
Classical Golden Mean,
Fibonacci Numbers,
and Platonic Solids
Alexey Stakhov
THE MATHEMATICS OF HARMONY
2
Chapter 1
The Golden Section
1.1. Geometric Definition of the Golden Section
1.1.1. A Problem of the Division in the Extreme and Mean Ratio
(DEMR)
The Elements of Euclid is one of the best known mathematical works of
ancient science. Written by Euclid in the 3rd century B.C., it contains the
main theories of ancient mathematics: elementary geometry, number theory,
algebra, the theory of proportions and ratios, methods of calculations of areas
and volumes, etc. Euclid, in this work, systematized a 300year period of de
velopment of Greek mathematics, and this work created a strong base for the
further development of mathematics. The information about Euclid himself is
extremely scanty. Except for several jokes, we only know that he taught at the
mathematical school in Alexandria. The Elements of Euclid surpassed all works
of his predecessors in the field of geometry, and during more than two millen
nia, The Elements remained the basic work for the teaching of “Elementary
Mathematics.” The 13 books of The Elements are dedicated to the knowledge
of geometry and arithmetic of the Euclidean epoch.
From The Elements of Euclid, the following geometrical problem,
which was named the problem of “Division in Extreme and Mean Ratio”
(DEMR), came to us [40]. This problem was formulated in Book II of
The Elements as follows:
Theorem II.11 (the area formulation of DEMR). To divide a line AB into
two segments, a larger one AC and a smaller one CB so that:
S ( AC ) = R ( AB,CB ) .
(1.1)
Note that S ( AC ) is the area of a square with a side AC and R ( AB,CB ) is
the area of a rectangle with sides AB and CB.
Chapter 1
The Golden Section
3
We can rewrite the expression
(1.1) in the following form:
S(AC)
( AC )2 = AB ×CB.
(1.2)
R(AB, BC)
Now, divide both parts of the
expression (1.2) by AC and then
by CB. The expression (1.2) then
takes the form of the following
proportion:
AB AC
=
.
(1.3)
A
C
B
AC CB
This form is wellknown to
Figure 1.1. A geometrical interpretation of
Theorem II.11 (The Elements of Euclid)
mathematicians as the “golden
section.”
We can interpret a proportion (1.3) geometrically (Fig. 1.2): divide a line
AB at the point C into two segments, a larger one AC and a smaller one CB, so
that the ratio of the larger segment AC to the smaller segment CB is equal to
the ratio of the line AB to the larger segment AC.
A
C
B
Figure 1.2. The division of a line in extreme and mean ratio (the golden section)
Denote a proportion (1.3) by x. Then, taking into consideration that
AB = AC + CB , the proportion (1.3) can be written in the following form:
AB AC + CB
CB
1
1
=
= 1+
= 1+
= 1+ ,
AC
x
AC
AC
AC
CB
whence we obtain the following algebraic equation:
x=
(1.4)
x 2 = x + 1.
It follows from the “geometrical meaning” of the proportion (1.3) that the
required solution of Eq. (1.4) has to be a positive number; it also follows that
a positive root of Eq. (1.4) is a solution of the problem. If we denote this root
by τ, then we obtain:
1+ 5
(1.5)
.
2
This number is called the Golden Proportion, Golden Mean, Golden Num
ber, or Golden Ratio.
τ=
Alexey Stakhov
THE MATHEMATICS OF HARMONY
4
We can write the following identity, which connects the powers of the
golden ratio:
(1.6)
τn = τn −1 + τn −2 = τ × τn −1 .
The approximate value of the golden proportion is:
τ ≈ 1.618033988749894848204586834365638117720309180….
Do not be astonished by this number! Do not forget that this number is an
irrational one! In this book, we will use the following approximate value:
τ ≈ 1.618 or even τ ≈ 1.62 .
This surprising number, which possesses unique algebraic and geometrical
properties, became an aesthetic canon of ancient Greek art and Renaissance art.
Why did Euclid formulate Theorem II.11? As is shown in [40], by using this
theorem, Euclid introduced the geometric construction of the golden triangle,
pentagram, and dodecahedron (these geometric figures will be discussed below).
1.1.2. The Origin of the Concept and Title of the Golden Section
Authorities vary over who introduced both the concept and terminology for
the golden section. According to [40], the concept of the DEMR (or the golden
section) was introduced in The Elements (see Theorem II.11). However, The Ele
ments is not a completely original work. There is an opinion that the majority of
theorems presented in The Elements are scientific results obtained by the Pythagore
ans. Roger HerzFischler wrote [40]: “Many authors point to references according
to which the pentagram was the symbol of the Pythagoreans, and from this, they
deduce that the Pythagoreans were probably acquainted with DEMR.” Herz
Fischler notes that the famous historians of mathematics, Heath and Van der Waer
den, supported this point of view. However, Euclid did not use the term “golden
section” in his works; instead, he used the term DEMR. In [40], we can trace the
history of terminology for the DEMR throughout the ages. As follows from [40],
the terminology of DEMR was used by Euclid, Fibonacci (1220), Zumberti (1516),
Gryaneus (1533), Candalla (1566), Billingsley (1570), Commandino (1572), Clavius
(1579), and Barrow (1722). However, many mathematicians used the term “Mid
dle and Two Ends,” in particular: Abu Kamil (850930), alBiruni (9731050), Ger
hard of Cremona (12th century), Adelard (12th century), Campanus of Novara
(13th century), and Billingsley (1570). In addition to the definitions of the “divi
sion in the extreme and mean ratio” and “proportion having a middle and two ends,”
other definitions, namely: “divina proportione,” “proportionally divided,” “contin
uous proportion,” “medial section,” “the golden number,” and “the golden section”
were used. The term “divina proportione” was the title of Pacioli’s book published
in 1509. This term was used by Kepler in his 1608 letter. We can find the term
Chapter 1
The Golden Section
5
“proportionally divided” in Clavius’ 1574 edition of The Elements. Kepler, in his
Mysterium Cosmographycum used the term “proportional division” for the DEMR.
The term “continuous proportion” was used by Euclid and we can also find it used
in Kepler’s 1597 letter. The term “medial section” was used by Leslie (1820) in
connection with the DEMR.
The “golden number,” the “golden section,” or the “golden mean” are the
most popular names for the DEMR. As is emphasized in [40], Tannery (1882)
used the term “Section d’Or” and Cantor (1894) used a title of the “golden
shnitt.” Also noted in [40], for the first time, the term “golden schnitt” (the
“golden section”) was used by Ohm in the second edition of his book Pure
Elementary Mathematics (1835).
However, other points of view exist about the origin of the term “the golden
section” — mainly in Russian literature. Edward Soroko, who is one of the most
authoritative Slavic researchers of the “golden section,” wrote in his book [25]:
“The title of the “Golden Section” (“Sectio Aurea”) takes its origin from
Claudia Ptolemey who gave this title for the number 0.618, after he had been
convinced that the growth of a personwith perfect constitution is divided
naturally in this ratio. The given title was fixed and then became popular due
to Leonardo da Vinci who often used this title.”
Unfortunately, the claim about the involvement of Leonardo da Vinci to
the introduction of the title of the golden section has been questioned be
cause there are no direct references to this title in his works. However, we
should not forget that the great Italian mathematician Luca Pacioli was first
to introduce the title “divine proportion” for the DEMR. Leonardo da Vinci
actively participated in Pacioli’s book De Divina Proportione as he illustrated
this unique mathematical work — which was the first book on the golden mean
in world history. This means that, without any doubt, Leonardo da Vinci knew
the concept of the “divine proportion.”
Very often, the golden mean is denoted by the Greek letter F (the number
PHI). This letter is the first one in the name of the wellknown Greek sculp
tor Phidias, who widely used the golden section in his sculptural works. Re
member that Phidias (5th century B.C.), together with Polyclitus, were con
sidered to be two of the most famous and authoritative masters of ancient
Greek sculpture of the Classical epoch. He became famous thanks to his su
pervision over designing the Acropolis. He created the enormous bronze statue
of Athena (“Winner in Battle”) in commemoration of the victory over the
Persians. Also, he created two grandiose statues from gold and ivory: Athena
Parthenos (“Maidens”) for the Parthenon and Zeus for the Olympian temple
of Zeus (about 430 B.C.), which is considered one of the “Seven Miracles of
Alexey Stakhov
THE MATHEMATICS OF HARMONY
6
the World.” Despite the monumental character of his sculptures — which
were unprecedented by their sizes among all Greek sculptures of that time
(for example, the 9meter sculpture of Athena Parthenos or the 13meter
sculpture of the Olympian Zeus) — they were constructed with strict steadi
ness and harmony based on the golden mean.
1.1.3. A Way of Geometrical Construction of the Golden Section
C
The golden section
arises often in geometry.
From the Euclidean Ele
D
ments, we know the fol
lowing form of geometric
construction of the gold
en section by using only a
pair of compasses and a
A
B
ruler (Fig. 1.3).
E
Construct a right tri
angle ABC with the sides Figure 1.3. The geometric construction of the golden section
AB=1 and AC=1/2. Then,
2
according to the Pythagorean Theorem, we have: CB = 1 + (1 / 2) = 5 / 2 .
By drawing the arc AD with the center at the point C before its intersec
tion with the segment CB at the point D, we obtain the segment
(
)
BD = CB − CD = 5 − 1 / 2 = τ−1 .
(1.7)
By drawing the arc DE with the center at the point B before its intersection
with the segment AB at the point E, we obtain a division of the segment AB at
the point E by the golden section because
AB EB
(1.8)
=
= τ or AB = 1 = EB + AE = τ −1 + τ −2 .
EB AE
Thus, the simple right triangle with leg ratio of 1:2, well known to ancient
science, could be the basis for the discoveries of the Pythagorean Theorem, the
Golden Section and Incommensurable Line Segments — the three great mathe
matical discoveries attributed to Pythagoras.
1.1.4. A “Double” Square
Many mathematical regularities, we can say, did “lie on the surface,” and
they needed only to be seen by a person with analytical, logical thinking that
Chapter 1
The Golden Section
7
was inherent in antique philosophers and mathematicians. It is possible that
ancient mathematicians could find the golden proportion by investigating
the socalled elementary rectangle with the side ratio of 1:2. This is named a
“double” or “twoadjacent” square because it consists of two squares (Fig. 1.4).
Let AB = 1 and AD = 2 . Then, if we
A
D
calculate the diagonal AC of the “dou
ble” square, then according to the
Pythagorean Theorem we get:
5
AC = 5.
If we take the ratio of the sum
AB + AC relative to the larger side АD
B
C of the “double” square, we come to the
Figure 1.4. A rectangle with a side ratio golden mean, because:
of 2:1 (a “double” square)
(
)
(1.9)
(AB+AC)/AD = 1 + 5 / 2 .
It is paradoxical that the Pythagorean Theorem is very wellknown to
each schoolboy while the golden mean is familiar to very few. The main pur
pose of this book is to tell about this wonderful discovery of antique science to
everyone who preserved the feeling to be surprised and admired, and we will
show to modern scientists the far, not trivial, applications of the golden mean
in many fields of modern science. We will tell our readers about this unique
mathematical discovery, which during the past millenia has attracted the at
tention of outstanding scientists, mathematicians and philosophers of the past,
like: Pythagoras, Plato, Euclid, Leonardo da Vinci, Luca Pacioli, Johannes
Kepler, Zeising, Florensky, Ghyka, Corbusier, Eisenstein, American mathe
matician Verner Hogatt — founder of the Fibonacci Association, and the Great
scientist Alan Turing — a founder of modern computer science.
1.2. Algebraic Properties of the Golden Mean
1.2.1. Remarkable Identities of the Golden Mean
What a “miracle” of nature and mathematics is the golden mean? Why
does our interest in it not wither, but, on the contrary, increases with each
century? To ponder this issue, we suggest that our readers focus all of their
mathematical knowledge and plunge into this fascinating aspect of the world
of mathematics. Only then there is the possibility of enjoying and under
THE MATHEMATICS OF HARMONY
Alexey Stakhov
8
standing the wonderful mathematical properties, beauty and harmony of this
unique phenomenon — the golden mean.
Let us start from the algebraic properties of the golden mean. It follows
from (1.4) — which is very simple and nevertheless a rather surprising property
of the golden mean. If we substitute the root τ (the golden mean) for x in Eq.
(1.4), then we will get the following remarkable identity for the golden mean:
τ2 = τ + 1.
(1.10)
Let us be convinced that the identity (1.10) is valid. For this purpose, it is
necessary to carry out elementary mathematical transformations over the left
hand and righthand parts of the identity (1.10) and to prove that they coincide.
In fact, we have for the righthand part:
1+ 5
3+ 5
τ +1 =
+1 =
.
2
2
On the other hand,
2
 1+ 5  1+ 2 5 + 5 3 + 5
2
=
=
,
τ =
 2 
4
2


from whence the validity of the identity (1.10) follows.
If we divide all terms of the identity (1.10) by τ we come to the following
expression for τ:
1
τ = 1+ .
(1.11)
τ
This can be represented in the following form:
1
τ −1 = .
(1.12)
τ
Now, analyze the identity (1.12). It is well known that any number а has
its own inverse number 1/a . For example, the fraction 0.1 is an inverse num
ber to 10. A traditional algorithm to get the inverse number 1/a from the
initial number а consists of the division of the number 1 by the number а. In
general case, this is a very complex procedure. Try, for example, to get the
inverse number of a = 357821093572. This can be fulfilled only by the use of
modern computer.
Consider the golden mean τ = 1 + 5 / 2, which is an irrational number.
How can we get the inverse number 1/τ? The formula (1.12) gives a very
simple answer to this question. To solve this problem, it is enough to subtract
1 from the golden mean τ. In fact, on the one hand,
1
2
2(1 − 5 )
2(1 − 5 )
5 −1
=
=
= 2
=
.
2
2
τ 1 + 5 (1 + 5 )(1 − 5 ) (1) − ( 5 )
(
)
Chapter 1
The Golden Section
9
While on the other hand, as follows from (1.12), the inverse number 1/τ
can be found in the following manner:
1
1+ 5
5 −1
= τ −1 =
−1 =
.
τ
2
2
However, we will get greater “aesthetic pleasure” if we carry out the
following transformations of the identity (1.10). Multiply both parts of the
identity (1.10) by τ, and then divide them by τ 2. The outcome we get is two
new identities:
(1.13)
τ3 = τ2 + τ
and
τ = 1 + τ −1.
(1.14)
If we continue to multiply both parts of the identity (1.13) by τ, then
divide both parts of the identity (1.14) by τ 2, and continue this procedure ad
infinitum, we will come to the following graceful identity that connects the
adjacent degrees of the golden mean:
(1.15)
τn = τn −1 + τn −2 ,
where n is an integer taking its values from the set {0,±1,±2,±3,…}.
The identity (1.15) can be expressed verbally as follows: “Any member of the
golden series (golden powers) is the sum of the previous two golden powers.”
This property of the golden mean is truly unique! It is very difficult to
believe that the following identity is “absolutely true”:
100
99
98
100
99
97
⎛ 1+ 5 ⎞
⎛ 1+ 5 ⎞
⎛ 1+ 5 ⎞
τ =⎜
+⎜
=⎜
(1.16)
⎟
⎟
⎜ 2 ⎟⎟ ,
⎜ 2 ⎟
⎜ 2 ⎟
⎠
⎠
⎝
⎠
⎝
⎝
however, its validity follows from the validity of the general identity (1.15).
Moreover, the following identity is also true:
100
96
⎛ 1+ 5 ⎞
⎛ 1+ 5 ⎞
⎛ 1+ 5 ⎞
⎛ 1+ 5 ⎞
+⎜
+⎜
=⎜
τ =⎜
(1.17)
⎜ 2 ⎟⎟ .
⎜ 2 ⎟⎟
⎜ 2 ⎟⎟
⎜ 2 ⎟⎟
⎠
⎠
⎝
⎠
⎝
⎠
⎝
⎝
There are an infinite number of identities similar to (1.17), and all of them
follow from the general identity (1.15).
100
1.2.2. The “Golden” Geometric Progression
Consider the following sequence of golden mean degrees:
− n −1
..., τ− n , τ ( ) ,..., τ−2 , τ−1 , τ0 = 1, τ1 , τ2 ,..., τn −1 , τn ,... .
{
}
(1.18)
The sequence (1.18) has very interesting properties. On the one hand, the
sequence (1.18) is a geometric progression because each of its element is
Alexey Stakhov
THE MATHEMATICS OF HARMONY
10
equal to the preceding one multiplied by the number τ, a Denominator or
Constant Multiplier in the geometric progression (1.18), that is,
τn = τ × τn −1 .
(1.19)
On the other hand, according to (1.15), the sequence (1.18) has the prop
erty of “additivity” because each element is equal to the sum of the two pre
ceding elements. Note, that the property (1.15) is characteristic only for the
geometrical progression with the denominator τ, and such geometrical pro
gression is named the Golden Progression, Golden Series or Golden Powers.
In this connection, any Logarithmic Spiral corresponds to a certain geo
metrical progression of the type (1.18), the opinion of many researchers is
that the property (1.15) distinguishes the golden progression (1.18) among
other geometrical progressions, and this fact is a reason for the widespread
prevalence of the golden logarithmic spiral in forms and structures of nature.
1.2.3. A Representation of the Golden Mean in the Form of Continued Fraction
Let us now prove another astonishing property of the golden mean based on the
identity (1.11). If we substitute the expression 1+1/τ for t in the righthand part of
(1.11), then we get the representation t in the form of the following continued fraction
1
τ = 1+
.
1
1+
τ
If we continue such substitution ad infinitum, we get the continued frac
tion of the following form:
1
(1.20)
τ = 1+
.
1
1+
1
1+
1
1+
1 + ...
A representation of (1.20) in mathematics is called a Continued or Chain
fraction. Note that the theory of continued fractions is one of the most impor
tant topics of modern mathematics.
1.2.4. A Representation of the Golden Mean in “Radicals”
Consider the identity (1.10). If we extract a square root from the right
hand and lefthand parts of the identity (1.10), then we will get the following
representation for τ:
τ = 1 + τ.
(1.21)
Chapter 1
The Golden Section
11
If we substitute the expression 1+ τ for τ in the righthand part of (1.21),
then we get the following representation for τ:
(1.22)
τ = 1+ 1+ τ .
If we continue such substitution ad infinitum, we will get another re
markable representation of the golden mean τ in “radicals”:
τ = 1 + 1 + 1 + 1 + ... .
(1.23)
Every mathematician intuitively aspires to express mathematical results
in the simplest and most compact form, and if he discovers such an expression,
it gives him “aesthetic pleasure.” In this respect, mathematical creative work
(aspiration for “aesthetic” expression of mathematical results), is similar to
the creative activity of a composer or poet, because their main aim is to find
perfect musical or poetic forms that give rise to aesthetic pleasure. Note that
the formulas (1.20) and (1.23) give us an aesthetic pleasure, arousing a feeling
of rhythm and harmony when we begin to think of the infinite repeatability of
the same simple mathematical elements in formulas for τ.
1.3. The Algebraic Equation of the Golden Mean
1.3.1. Polynomials and Equations
Since ancient times, mathematicians paid special attention to the study of
polynomials and the solutions of algebraic equations, and this important math
ematical problem promoted the development of algebra. As is well known, a
Polynomial of nth degree is represented by the following expression:
(1.24)
an x n + ... + a2 x 2 + a1 x + a0 , where
In other words, a polynomial is the sum of the integervalued degrees taken
with certain coefficients. For example, a decimal notation of a number is, in essence,
some representation of this number in the form of a polynomial of the number 10,
for instance, 365 = 3 × 102 + 6 × (10 ) + 5 . If x is a variable, then a polynomial gives
any Polynomial Function, the range of which coincides with the range of the variable
x. The polynomials of the first, second, third, and fourth degrees are called Linear,
Quadratic, Cubic, and Biquadrate polynomials, respectively. The Algebraic Equation
(in the standard form) is a statement written with algebraic designations that some
polynomial function is equal to zero for some values of a variable x. For example,
( )
Alexey Stakhov
THE MATHEMATICS OF HARMONY
12
x 2 − 5 x + 6 = 0 is an algebraic equation. The values of the variable, for which the
polynomial becomes equal to 0, are named roots of this polynomial. For example,
the polynomial x 2 − 5 x + 6 has two roots, 2 and 3, because 22 − 5 × 2 + 6 = 0 and
32 − 5 × 3 + 6 = 0. Note that in the polynomial x 2 − 5 x + 6 the variable x represents
an arbitrary number from the range of the given polynomial function, but the alge
braic equation x 2 − 5 x + 6 = 0 , in contrast, means that the variable x can only be
the number 2 or 3, the roots of this algebraic equation.
Algebraic equations have always served as a powerful means to solutions of
practical problems. The exact language of mathematics has allowed mathemati
cians to simply express the facts and relations, which, when presented in ordi
nary language, can seem confusing and complicated. The unknown values, de
noted by some algebraic symbols, for example x, can be found if we formulate
the problem in mathematical language through the form of equations. Methods
of an equation’s solution are the basis for the Equation Theory in mathematics.
Now, recall the basic data of linear and square algebraic equations that are
wellknown to us from secondary school. The linear equation, in general form,
can be written as ax + b = 0, where a and b are some numbers and a ≠ 0. This
equation has one solution: x = −b / a ; that is, the linear equation has only one
root. The quadratic equation has the following form:
(1.25)
ax 2 + bx + c = 0, where a ≠ 0.
The rules for the solution of algebraic equations of the first and second
degree were wellknown in antiquity. For example, as we recall from second
ary school mathematics, we know the following formula for the roots of the
quadratic equation (1.25):
−b ± b2 − 4ac
(1.26)
.
2a
Remember that the values of the roots depend on the Determinant D of
the quadratic equation (1.25) — which is defined by the following formula:
x1,2 =
D = b2 − 4ac.
(1.27)
If the determinant D is positive, the formula (1.26) gives exactly two real
roots. If D = 0, then x = −b / 2a , and we say that Eq. (1.25) has two equal roots.
If the determinant D is negative, we have to introduce an imaginary unit i, which
is defined by i = −1, and for this case, both of these roots are complex.
1.3.2. Quadratic Irrationals
The discovery of irrational numbers is the greatest mathematical discovery of
Greek mathematics. This discovery was made by Pythagoreans in the 5th century
Chapter 1
The Golden Section
13
B.C. during the investigation of the ratio of two geometric segments: a diagonal and a
side of a square. By using Reductio ad Absurdum, that is, the Indirect Method of Con
tradiction, the Pythagoreans proved that the ratio of the diagonal to the side of the
square, which is equal to 2 , cannot be represented in the form of the ratio of two
natural numbers. Such geometric segments were named Incommensurable, and the
numbers, which express similar ratios, were named Irrational Numbers.
The discovery of Incommensurable Segments resulted in the first crisis of
the foundations of mathematics, and, ultimately, became a turning point in
the development of mathematics. One legend says that in honor of this dis
covery, Pythagoras had carried out a “hecatomb,” namely, he had sacrificed
100 bulls to the gods. In the beginning, the Pythagoreans attempted to keep
the new discovery a secret. According to legend, Hippias, one of the Pythagore
ans, broke the oath and divulged the secret of this discovery. Later, he per
ished during a shipwreck that the Pythagoreans considered to be a punish
ment from the gods for the disclosure the discovery’s secret. The influence of
this discovery on the development of science can be compared to the discov
ery of Lobachevsky’s geometry in the first half of the 19th century, and with
the discovery of Einstein’s Relativity Theory at the beginning of the 20th
century. Thanks to this discovery, mathematics received an entirely new math
ematical concept — the concept of Irrational Numbers.
The title “rational” is from the Latin word “ratio,” which is a translation of
the Greek word “logos.” The numbers, which can be represented as the ratios
of two integers, were called “rational.” In contrast to rational numbers, those
numbers, which express the ratios of the incommensurable line segments, had
been named irrational (from the Greek word “alogos”).
Unlike a linear equation, a quadratic equation of the kind (1.25) with ra
tional coefficients can have irrational roots called quadratic irrationals. Book
10 of The Elements is devoted to the study of the quadratic irrationals.
Scientists of India, the Middle East, and many European mathematicians of
the Middle Ages studied quadratic irrationals. It was proved that any number
of the kind N for any integer N, which is not a full square, is an irrational
number as well as a number 3 N , where N is not a cube, etc. As they say, similar
irrationals can be represented in Radicals. In 16th century, Italian mathemati
cians Tartalja, Cardano, and Ferrari found a general formula for the roots of the
cubic and biquadrate algebraic equations that represented their irrational roots
in radicals. Despite the fact that attempts to find general formulas for the roots
of the algebraic equations of the fifth and more degrees appeared unsuccessful,
this research direction resulted in new mathematical discoveries, which were
connected with the mathematical works of Niels Abel and Evariste Galois.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
14
1.3.3. Poor “Studious” Niels Abel
The history of science and mathematics contains many tragic
pages. One of them is the life and scientific work of the Norwe
gian mathematician Niels Abel, whose mathematical research in
volves algebra and solutions to algebraic equations.
For a long time, mathematicians assumed that an irratio
nal root of any algebraic equation with rational factors can
be expressed in radicals. However, in the 19th century, Abel
proved that the irrational roots of the general equations of Niels Henrik Abel
(18021829)
the 5th and more degrees cannot be expressed in radicals.
On the big square in the city of Oslo, the capital of Norway, a majestic
monument rises up above the cityscape. On the granite stone of the monu
ment, a young man is carved with a spiritualized face struggling against two
disgusting monsters. The monument is of the wellknown Norwegian mathe
matician Niels Henrik Abel. What do these monsters symbolize? Some math
ematicians joke that these monsters represent the equations of the fifth de
gree and the elliptic functions conquered by Abel. Others consider that it is
allegory: the sculptor wished to embody the social injustice with which Abel
struggled all his life.
The sad life of this Norwegian mathematical genius is typical for scientific
geniusesnot just from his country or of his time. Unfortunately, the destiny
of many mathematical geniuses unfolds tragically. It is connected with the
fact that many of great mathematicians are often not understood, and their
stunning mathematical discoveries are not recognized by their contemporar
ies, unfortunately, they are only recognized some time after their death.
Abel was born in 1802 in Northwest Norway in the small fishing town of
Finnej, where there were neither mathematicians, nor mathematical textbooks
available to him. Very little is known about the first years of his childhood. He was
enrolled in school in Oslo at the age of thirteen. Abel studied lightly and obtained
good marks excelling in mathematics. He also liked to play chess and visit the
theatre. However, within three years of school, a sudden change opened up his
world. Instead of a cruel mathematics teacher who beat pupils, the new mathe
matics teacher, Holmboe, arrived at the school. He knew his subject very well,
and he spiked the interest of his pupils. Holmboe gave each pupil the opportunity
to act independently, and encouraged them to take the first steps paid special
attention towards a more fulfilling mathematics education. Soon, Abel not only
took a great interest in mathematics, but also found that he was capable of solv
ing mathematical problems of which other people were incapable.
Chapter 1
The Golden Section
15
Abel’s family lived in humiliating pov
erty so he was attending a school free of
charge. In 1820, Abel’s father died, and the
family remained without any financial
means. Their situation was desperate. Niels
thought about returning to his native city
and searching for a job, but some of his pro
fessors paid attention to the young man’s
talent and helped him to enroll in the local
university. They managed to obtain grants
for Niels Abel to preserve his unique math
ematical talent in science, and for travel
ing abroad. Abel’s visits to Berlin, Paris, and
to other large mathematical centers of that
time contributed greatly to his future
mathematical works. Unfortunately, the
young mathematician’s discoveries which
A monument of Niels Abel in Oslo
surpassed much the science of the time resulted in a lack of understanding and un
derestimation by his contemporaries. Abroad, as well as in his native land, Abel ex
perienced great financial difficulties and a constant feeling of intolerable loneliness.
His attempts to get scientific recognition were unsuccessful. He sent his works to
the Parisian Academy where they were forwarded to the French mathematician,
Cauchy. Unfortunately, these works were lost. Also his letter to the German math
ematician Gauss was left unanswered.
The young mathematician, who made a revolution in mathematics, re
turned home to remain the same poor and unknown “studious” Abel. He
could not find a job. Being ill by tuberculosis, and “poor as a church mouse,”
by his own words, Abel, in a condition of the cheerless melancholy, died on
April 6, 1829, at 26.
The proof of insolvability in radicals of the algebraic equations of the
fifth and greater degrees was his most important mathematical discov
ery, made at just 22. In the opinion of the mathematician Ermit, “Abel
left such a rich heritage for mathematicians that they will develop it dur
ing the next 500 years.”
There is a custom, to which new results and discoveries are named after
the scientists who made them. Now, anyone holding a book on higher mathe
matics can see that Abel’s name is immortalized in various areas of mathemat
ics. There are several theorems and mathematical results bearing Abel’s name:
Alexey Stakhov
THE MATHEMATICS OF HARMONY
16
Abelian integrals, Abelian equations, Abelian groups, Abelian formulas, and
Abelian transformations. Abel would be surprised to learn that his work has had
a profound influence upon the development of mathematics.
1.3.4. Mathematician and Revolutionist Evariste Galois
The destiny of one more genius, the French mathematician Evariste Galois,
is not less tragic. He was born on October 26, 1811, in the small town of Bourg
laReine near Paris, France.
For the first twelve years of his life his mother, who was a
fluent reader of Latin and classical literature, was responsible
for his education. In October 1823, he entered the Lycee Lou
isleGrand. At LouisleGrand, Galois enrolled in the mathe
matics class of Louis Richard. He worked more and more on
his own researches and less and less on his schoolwork. He stud
ied Legendre’s Geometrie and the treatises of Lagrange. His
teacher Richard reported, “This student works only in the
highest realms of mathematics.” In April 1829 Galois published
his first mathematics paper on continued fractions in the An
nales de Mathematiques. Later he submitted articles on the
Evariste Galois
(18111832)
algebraic solution of equations to the Academie des Sciences.
Cauchy was appointed as referee of Galois’ paper. On Decem
ber 29, 1829 Galois has received his Baccalaureate degree. In the development of
Abel’s work that he learned from Bulletin de Ferussac, and by the Cauchy’s advice,
he submitted a new article On the Condition that an Equation be Soluble by Radi
cals in February 1830. The paper was sent to Fourier, the secretary of the Paris
Academy, to be considered for the Grand Prize in mathematics. Since Fourier died
in April 1830, Galois’ paper was never found and considered for the prize. Further
Galois’ works on the theory of elliptic functions and Abelian integrals were initiated
by Abel and Jacobi’s works. With support from Jacques Sturm, he published three
papers in Bulletin de Ferussac in April 1830.
Galois enthusiastically participated in revolutionary activity, and even
tually found himself in a prison for several times. In March 1832, being in a
prison, he felt ill with cholera and was sent to the pension Sieur Faultrier
where he fell in love with StephanieFelice du Motel, the daughter of the
resident physician. In May 1832 his stormy life finished: he was killed in a
duel. The reason for the duel was not clear but certainly linked with
Stephanie. Before the duel, he wrote a resume of his mathematical discover
ies and transferred his notes to one of his friends, requesting him to pass it on
Chapter 1
The Golden Section
17
to one of the leading mathematicians. The note ended with the following
words: “You will publicly ask Jacobi or Gauss to give a conclusion not about
validity, but about the value of these theorems. Then, I hope, people, who
can decipher all this mess, will be found.” From what is known, Galois’ letter
reached neither Jacobi nor Gauss. The mathematical community only learned
of Galois’ works in 1846, when the French mathematician Liouville published
most of Galois’ works in his mathematical journal. All of Galois’ works were
presented in a small format of 60 pages. These works contained a presenta
tion of group theory — now considered to be the “key” theory of modern
algebra and geometry, the first classification of irrationals defined by alge
braic equations. This doctrine is now referred to as Galois Theory.
1.3.5. The “Golden”Algebraic Equations of nth Degree
And now, after such fundamental mathematical training, we again turn to
the equation of the golden proportion (1.4). Clearly, it is the quadratic algebraic
equation of the type (1.25) with the factors:
(1.28)
a = 1; b = −1; c = −1.
By using (1.27) and (1.28), we can calculate the determinant of Eq. (1.4):
D = 5.
It follows from here that Eq. (1.4) has two real roots:
1+ 5
1 1− 5
.
and x2 = − =
(1.29)
2
τ
2
The root x1 coincides with the golden mean τ.
Typically, the main problem of the theory of algebraic equations is to
find their roots. We know the elementary algebraic equation (1.4) with the
root equal to the golden mean. With this equation, the following question
arises: are there algebraic equations of the highest degrees with the root equal
to the golden mean? And if so, what do they look like? To answer this ques
tion we will use the following reasoning concerning the initial equation of
the golden mean (1.4).
Multiply both parts of Eq. (1.4) by x; as a result we get the following
equality:
x1 = τ =
x 3 = x 2 + x.
(1.30)
Equation (1.4) can be written in the following form:
x = x 2 − 1.
If we substitute x2 1 for x in the expression (1.30), we get the following
THE MATHEMATICS OF HARMONY
Alexey Stakhov
18
algebraic equation of the third degree:
(1.31)
x 3 = 2 x 2 − 1.
2
On the other hand, if we substitute x +1 for x in (1.30), then we get one
more equation of the third degree:
(1.32)
x 3 = 2 x + 1.
Thus, we have obtained two new algebraic equations of the third de
gree. Now, prove that the golden mean is a root of Eq. (1.31). With this
end, we substitute the golden mean τ = 1 + 5 / 2 for x to the lefthand
and righthand parts of Eq. (1.31) and show that both parts of this equa
tion coincide. In fact, by using the identity (1.15), we can obtain for the
lefthand part:
(
)
3 + 5 1+ 5 4 + 2 5
+
=
= 2 + 5.
2
2
2
We can also obtain for the righthand part of this equation:
τ3 = τ2 + τ =
3+ 5
− 1 = 2 + 5.
2
Hence, the golden mean τ is, in fact, a root of Eq. (1.31), that is, Eq.
(1.31) is a golden one. By analogy, we can prove that Eq. (1.32) is a golden
one as well.
Let us derive the golden algebraic equation of the fourth degree, for this
purpose we multiply both parts of the equality (1.30) by x; as a result we will
get the following equality:
2τ 2 − 1 = 2 ×
(1.33)
x4 = x3 + x2.
2
We then use the expression (1.4) for x and the expressions (1.31) or (1.32)
3
for x . If we substitute them into the expression (1.33), we obtain two new
golden algebraic equations of the fourth degree:
x 4 = 3x 2 −1
(1.34)
(1.35)
x 4 = 3 x + 2.
The analysis of Eq. (1.35) resulted in unexpected result. It is found, that
this equation describes the energetic state of the butadiene molecule, a valu
able chemical substance used in the production of rubber. American scientist
Richard Feynman, Nobel Prize Laureate, expressed his enthusiasm concern
ing Eq. (1.35) in the following words: “What miracles exist in mathematics!
According to my theory, the golden proportion of the ancient Greeks gives
the minimal energy condition of the butadiene molecule.”
This fact, at once, raises our interest in the golden equations of the higher
Chapter 1
The Golden Section
19
degrees, which, probably, describe energy conditions of the molecules of other
chemical substances. These equations can be obtained, if we will consistently
consider the equalities of the kind x n = x n −1 + x n −2 . As an example, we can derive
the following golden equations of the higher degrees:
x 5 = 5 x 2 − 2 = 5 x + 3;
x 6 = 8 x 2 − 3 = 8 x + 5;
x 6 = 13 x 2 − 5 = 13 x + 8.
The analysis of these equations demonstrates that Fibonacci numbers (1, 1,
3, 5, 8, 13, 21, 34, 55) — which are addressed in Chapter 2 — are the numerical
factors in the righthand part of these equations. It is easy to show, that in the
general case, the golden algebraic equation of the nth degree can be expressed in
the following form:
x n = Fn x 2 − Fn −2 = Fn x + Fn −1 ,
(1.36)
where Fn , Fn −1 , Fn −2 are Fibonacci numbers.
Note, once again, that the main mathematical property of all equations of the
kind (1.36) is the fact that all of them have a common root — the golden mean.
Thus, our simple reasoning resulted in a small mathematical discovery:
we have found an infinite number of new golden algebraic equations given by
(1.36) — which have the golden mean as a common root.
If we substitute in Eq. (1.36) its root τ = 1 + 5 / 2 for x, then we obtain
the following remarkable identities that connect the golden mean with Fi
bonacci numbers:
(1.37)
τn = Fn τ2 − Fn −2 = Fn τ + Fn −1 .
Below we show that, for instance, the18th, 19th, and 20th Fibonacci num
bers are equal to the following:
(1.38)
F18 = 2584, F19 = 4181, F20 = 6765.
Then, taking into consideration (1.38) and using (1.36), we can write the
following algebraic equations of the 20th degree:
(1.39)
x 20 = 6765 x 2 − 2584
(
)
(1.40)
x 20 = 6765 x + 4181.
τ
=
1
+
5
/
2
is a root of
It is difficult to imagine that the golden mean
the equations (1.39) and (1.40). However, this fact follows from the theory of
golden algebraic equations given by (1.36), and by looking at the equations
(1.36), (1.39) and (1.40). Once again, we are convinced of the greatness of
mathematics — which allows us to express the complex scientific information
in such compact form.
(
)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
20
1.4. The Golden Rectangles and the Golden Brick
1.4.1. A Golden Rectangle with a Side Ratio of τ
Many different definitions A
of the golden rectangles exist.
First, we will study the golden
rectangle with a side ratio τ
(Fig. 1.5). A rectangle in Fig.
1.5 is a golden one because the
ratio of its larger side to the
smaller side is equal to the gold
en mean, that is:
E
B
O
G
I
H
1+ 5
F
D
C
.
Figure
1.5.
Golden
rectangle
with
a
side
ratio
of
τ
2
Consider the golden rectangle with the sides AB = τ and BC = 1 . We can
find points E and F on the line segments АВ and DC, which divide the seg
ments АВ and DC in the golden section.
It is clear that if AE = DF = 1 , then EB = AB − AE = τ − 1 = 1 / τ .
Now, we draw the line segment EF, which is called a Golden Line. It is
clear that the golden line EF divides the golden rectangle ABCD into two
new rectangles AEFD and EBCF. It follows from the above geometric con
siderations that the rectangle AEFD is a square.
Now, let us consider the rectangle EBCF. As its big side BC = 1 and its
small side EB = 1 / τ , hence their ratio BC : EB = τ . It means that the rectan
gle EBCF is a golden one also. Thus, the golden line EF divides the initial
golden rectangle ABCD into a square AEFD and a new golden rectangle EBCF.
Now, draw the diagonals DB and EC of the golden rectangles ABCD and
EBCF. It follows from the similarity of the triangles ABD, FEC, BCE that the
point G divides the diagonal DB in the golden section. Then, we draw a new
golden line GH in the golden rectangle EBCF. It is clear that the golden line GH
divides the golden rectangle EBCF into a square GHCF and a new golden rect
angle EBHG. By repeating this procedure infinitely, we will get an infinite se
quence of squares and golden rectangles converging in the limit to the point O.
Note that such endless repetition of the same geometric figures, that
is, squares and golden rectangles, gives rise to aesthetic feelings of harmo
ny and beauty. It is considered, that this fact is a reason why many rectan
AB : BC = τ =
Chapter 1
gular forms (match boxes, lighters, books,
suitcases) frequently are golden rectan
gles. Below we will discuss the applica
tions of the golden rectangle to architec
ture and painting.
For example, we widely use credit
cards in our daily life, however we do not
pay attention that in many cases our cred
it cards have a form that is, or approxi
mates to the golden rectangle (Fig. 1.6).
The Golden Section
21
Figure 1.6. A credit card has a form
of the golden rectangle
1.4.2. A Golden Rectangle with a Side Ratio of τ 2
We look back to Theorem II.11 of The Elements of Euclid. Consider, once
again, a division in the extreme and mean ratio represented in Fig. 1.3, and
according to Theorem II.11, we should construct the line segments АВ, АС,
and СВ, which form the golden mean (Fig. 1.2), a rectangle for the condition
(1.2). We name the rectangle a Euclidean rectangle. If we designate the lengths
of the line segments АВ, АС and СВ as AB = a, AC = b and CB = c, then, we
can rewrite the expression (1.2) as follows:
a × c = b2 .
(1.41)
Taking into account (1.41), we can represent
the Euclidean rectangle as is shown in Fig. 1.7.
c = τ2 = 0.382
It follows from Fig. 1.7 that a ratio of the
larger side to the smaller one in the Euclidean
rectangle is equal to the ratio of the initial seg
a =1
Figure 1.7. The Euclidean rectangle ment to the smaller segment in Theorem II.11
of The Elements here, according to (1.41), its
area is equal to a square of the larger segment.
If we take the segment of the length 1 as the initial segment ( a = 1) , then the
lengths of the larger segment (b) and smaller one (c) in the DEMR will always be
a proper fraction, and the expression (1.41) can be written in the following form:
c = b2 .
(1.42)
The following formulation of Theorem II.11 for the case of the line seg
ment of the length 1 comes directly from (1.42).
Theorem II.11 for the segment of the length 1. We divide the line seg
ment of the length 1 into two unequal parts in golden ratio; thus, the length of
the smaller line segment will be equal to the square of the larger line segment.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
22
If the initial segment АВ in Fig. 1.1 will be the line segment of length 1,
−1
that is, AB = 1 , then the larger segment AC = τ , and the smaller segment
CB = τ−2 . Taking into account this fact, we can say that the Euclidean rectan
gle in Fig. 1.7 is a new golden rectangle with a side ratio AB : CB = τ2 . It fol
lows from this consideration that Euclid, in his Theorem II.11, not only for
mulated the DEMR (the golden section), but he also discovered a new golden
rectangle with a side ratio τ2 .
Note that Theorem II.11, for the segment with length 1, presents the
following wellknown property of the golden mean:
1 = τ–1 + τ–2 = 0.618 + 0.382.
(1.43)
The identity (1.43) is a partial case of the identity (1.6), and it expresses
the famous “Principle of the Golden Proportion” that has been widely used
in human culture since antiquity.
1.4.3. The “Golden” Brick of Gothic Architecture
Again, we will use the Euclidean rectangle in Fig. 1.8, where the larger
side is AB = τ and the smaller side is AD = τ−1 . Draw the Euclidean rectangle
in Fig. 1.7 with a diagonal DB.
A
B
τ 1
3
τ
D
C
Figure 1.8. A calculation of the diagonal in the Euclidean rectangle
By using the Pythagorean Theorem, we can write:
DB = BC 2 + DC 2 = τ2 + τ–2.
Calculate numerical values for τ2 and τ−2 . In fact, we have:
2
(1.44)
2
1+ 5  1+ 2 5 + 5 3 + 5
=
=
τ =
.
 2 
4
2


On the other hand, we have:
2
(1.45)
2
τ
−2
 5 −1 5 − 2 5 +1 3 − 5
=
=
=
.
 2 
4
2


(1.46)
Chapter 1
The Golden Section
23
Taking into account (1.45) and (1.46), we can rewrite the expression (1.44)
as follows:
DB2 = 3.
(1.47)
It comes from (1.47) that
DB = 3.
(1.48)
As is shown in [26], the Euclidean rectangle in Fig. 1.8, together with the
classical golden rectangle in Fig. 1.5, can be used for the construction of a
special kind of rectangular parallelepiped — also called a Golden Brick (Fig.
1.9). Consider this parallelepiped for the case AB = DC = 1 .
The faces of the golden brick in
F
Fig. 1.9 are the golden rectan
gles with geometric ratios
G
based on the golden
2
mean. The face
1
ABCD is a clas
E
τ
sical golden
H
rectangle with A
3
the side ratio of
B
1
τ. This means
τ
that the edge is
AD = τ−1 . The
τ
face ABGF is D
also a classical
1
C
golden rectan
gle with the
Figure 1.9. The golden brick
side ratio of τ.
This means that the edge AF = τ . Finally, the rectangle BCHG is the Euclid
ean rectangle with the side ratio of τ2 (Fig. 1.7). This means that the edges
are BG = τ and BC = τ−1 . Note that the diagonal CG = 3.
By using the Pythagorean Theorem, we can calculate the diagonal CF of
the golden brick:
CF = FG 2 + CG 2 = 1 + ( 3 )2 = 2.
In the book [26], evidence is suggested that the golden bricks were wide
ly used in the Gothic castles as a form of basic building blocks. In [26], the
hypothesis is drawn that the surprising strength of the Gothic castles is bound
up with the use of the golden bricks during the construction of architectural
monuments of gothic style.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
24
1.5. Decagon: Connection of the Golden Mean to the Number π
The quantity of irrational (incommensurable) numbers is infinite. Howev
er, some of these numbers take a special place in the history of mathematics —
specifically, in the history of material and spiritual culture. Their significance
involves the fact that they express proportions and relations — which have uni
versal character — and often appear in the most unexpected places. The irratio
nal number 2 — which is equal to the ratio of the diagonal to the side of a
square — is the first of them. Recall that the discovery of the socalled “incom
mensurable line segments” is connected directly with this number. This discov
ery caused the most dramatic period in ancient mathematics. It resulted in the
development of the theory of irrational numbers and ultimately to the creation
of modern “continuous” mathematics.
Two mathematical constants — the number π, which expresses the ratio of
a circle length to its diameter, and “Euler’s Number” е, the base of natural
logarithms are the next important irrational numbers. Their significance
consists in the fact that they originate the main classes of the “elementary
functions”: trigonometric functions (the number π) and also exponential func
tion ex, logarithmic function logex, and lastly hyperbolic functions (the num
ber е). The numbers π and е, that is, two of the most important constants of
mathematical analysis, are connected by the following astonishing formula:
1 + e ix = 1,
(1.49)
where i = −1 is an imaginary unit.
This identity is not recognized by
everyone at first sight. However, in con
a10
sidering this identity, which connects
among themselves the fundamental
R
mathematical constants, the numbers
36°
π and e, it is difficult to refrain from
pondering the mystical character of
this mathematical formula (1.49).
The golden mean τ refers also to a
category of the fundamental mathe
Figure 1.10. Decagon
matical constants, but this poses a
question. Is there some connection be
tween these mathematical constants, for example, between the numbers π
and τ? The analysis of a regular Decagon (Fig. 1.10) gives the intriguing an
swer to this question.
Chapter 1
The Golden Section
25
Consider a circle with a radius R and a decagon inscribed inside the circle
(Fig. 1.9). It is known from geometry that the side a10 of the decagon is con
nected to the radius R by the following formula:
a10 = 2Rsin18°.
(1.50)
If we carry out some trigonometric transformations by using the for
mulas, wellknown from secondary school trigonometry, we obtain the fol
lowing outcomes:
(1) The side of the decagon inscribed inside the circle with radius R is
equal to a large part of the radius R divided in the golden section, that is:
a10 = R / τ.
(2) The golden mean τ is connected with the number π by the following
correlation:
τ = 2cos36° = 2cos(π/5).
(1.51)
This formula, which is obtained as an outcome of mathematical analysis of
the decagon, is one more confirmation of the fact that the golden mean to
gether with the number π, shares the status of being amongst main mathe
matical constants of nature.
Consider the following from the area of nuclear physics. By studying the
laws of the nuclear kernel, Byelorussian physicist Vasily Petrunenko, in the
book The Golden Section of Quantum States and its Astronomical and Physical
Manifestations (2005) [53], came to the following conclusion: a huge stabil
ity of the nuclear kernel is due to the wave’s multiplicities of the golden mean
underlying their organization. Thus, he proved that the regular decagon un
derlies the structure of the nuclear kernel!
1.6. The Golden Right Triangle and the Golden Ellipse
A
1.6.1. The Golden Right Triangle
The right triangle based upon the golden sec
tion is used widely in architecture. Consider the
right triangle АВC with the side ratio AC : CB = τ
(Fig. 1.11).
If we designate x, y, and z as the side lengths of the
right triangle АВС and also take into account that the
z
y
C
51°50'
x
B
Figure 1.11. The golden
right triangle
Alexey Stakhov
THE MATHEMATICS OF HARMONY
26
ratio y : x = τ , then according to the Pythagorean Theorem, the length z of the
side AB can be calculated by the formula:
z = x 2 + y2 .
(1.52)
If we take x = 1, y = τ , then we have:
z = 1 + τ = τ2 = τ.
The right triangle with the side ratios τ : τ :1 is called a Golden Right Tri
angle.
Below, we will show that the golden right triangle is the main geometric
idea of the Great Pyramid in Giza.
1.6.2. The “Golden” Ellipse
C
The golden ellipse [26] is
H
G
formed with the help of the two
golden rhombi (ACBD and ICJD)
inscribed into the ellipse (Fig.
J
1.12). The golden rhombi ACBD I
O
A
B
and ICJD consist of four right tri
angles of the kind OCB or OCJ,
which are the golden right trian
gles (Fig. 1.11).
F
E
D
Now, consider the basic geo
metric correlations of the golden
Figure 1.12. The golden ellipse
ellipse in Fig. 1.12. Let a focal dis
tance AB of the ellipse in Fig. 1.12 be equal to AB = 2 . The following correla
tion follows from the ellipse definition:
AC + CB = AG + GB.
(1.53)
The following correlations, which connect the sides of the golden
right triangles OCB and OCJ, follow from the definition of the golden
right triangle. As the smaller leg OB of the golden right triangle is equal
to 1, that is, OB = 1 , it follows from the definition of the golden right
triangle that the values for the hypotenuse and the larger leg of the
triangle OCB are:
CB = τ and OC = τ .
(1.54)
Let us consider the golden right triangle OCJ. Because its smaller leg
OC = τ, then, according to the definition of the golden right triangle, we have:
OJ = OC × τ = τ × τ = τ and CJ = τ × OC = τ × τ .
(1.55)
Chapter 1
The Golden Section
27
There is also the wellknown correlation that connects the halfaxes OI and
ОС with the line segment OB of the ellipse in Fig. 1.12:
(1.56)
OC 2 = OI 2 − OB2 .
We know that the correlation (1.56) is true for the figure in Fig. 1.11. In fact,
if we substitute in (1.56) the numerical values of the line segments
OI = τ, OC = τ and OB = 1, we can obtain:
2
τ = τ2 − 1 or τ = τ2 − 1
that is equivalent to the identity (1.6). This means that the correlation (1.56)
is valid and the figure in Fig. 1.11 is an ellipse.
In the opinion of the Polish scientist Jan Grzedzielski, author of the book
EnergeticznoGeometryczny Kod Phzyrody [26], the golden ellipse can be used
as a geometric model for the spreading of the light in optic crystals, that is, the
geometrical correlations of the golden ellipse give optimal conditions for the
attainability by photons of the focuses with minimal energetic losses.
( )
1.7. The “Golden” Isosceles Triangles and Pentagon
1.7.1. Construction of the “Golden” Isosceles Triangle and Regular
Pentagon in The Elements
Theorem II.11 played an important role in
A
C
B
The Elements by Euclid. Using this theorem, Eu
clid constructs the isosceles triangle and regu
lar pentagon, which are then used for the con
struction of the dodecahedron (see Chapter 3).
First, we construct an isosceles triangle whose
D
angles at its base are double relative to the ver
tex angle (Fig. 1.13). We can do this by drawing
the line AB and then divide it in the golden sec Figure. 1.13. A geometric construc
tion at the point C. Then, we draw a circle with tion of the golden isosceles triangle
the cenetr A and the radius AB. We mark on the circle the point D so that
AC=CD=BD. The triangle ABD has the property that its angles B and D, at the
base BD, are double relative to its vertex angle A. Note that the vertex angle
A=36° and the angles B=D=72°. As C is the golden section point, AC=CD=BD,
and AD=AD, this means that the ratio of the hips AB and AD to the base BD of the
isosceles triangle ABD is equal to the golden mean, that is:
Alexey Stakhov
THE MATHEMATICS OF HARMONY
28
AB AD
=
=τ
BD BD
The isosceles triangle with a golden ratio
(1.57) is aptly called the Golden Isosceles Tri
angle (Fig. 1.13).
By using the triangle ABD, we draw a cir
cle through A, B, and D (Fig. 1.14). Then we
bisect the angle ADB by the line DE, which
meets the circle at the point E. Note that the
line DE passes through the point C, which di
vides a line AB in the golden section. Similarly
we can find a point F and then we can draw the
pentagon AEBDF.
(1.57)
A
E
F
C
D
B
Figure 1.14. A geometric con
struction of a regular pentagon
1.7.2. A Regular Pentagon
Let us consider a regular pentagon presented in Fig. 1.15.
If we draw in the regular pentagon all di
A
agonals, we obtain a pentagonal star called a
Pentagram or Pentacle. Note that the word
“pentagon” originates from the Greek word
F
G
B pentagonon and the word “pentagram” origi
E
nates from the Greek word “pentagrammon”
H
(“pente” is five and “grammon” is a line). A
L
“pentagram” is a regular pentagon with gold
K
en isosceles triangles constructed on each side.
C
D
It is proved that the crossing points of diago
nals in the pentagon are always the points of
Figure 1.15. Regular pentagon
and pentagram
the golden section. Thus, they form a new reg
ular pentagon: FGHKL. In this new pentagon,
we can draw diagonals at their crossing points to form another pentagon.
This process can be continued infinitely. Thus, the pentagon ABCDE consists
of an infinite number of the pentagons formed by the crossing points of diag
onals. This infinite repetition of the same geometrical figures creates a feel
ing of rhythm and harmony, which is perceived by our intellect.
The regular pentagon in Fig. 1.15 is a rich source of golden sections. If we
assume that a side of the inner pentagon FGHKL is equal to 1, that is,
FG = GH = HK = KL = 1, then we get the following properties of the regular
pentagon and pentagram ABCDE based on the golden section:
Chapter 1
The Golden Section
29
1. All five triangles, AGF, BHG, CKH, DLK, and EFL, are golden isosceles
triangles; here the lengths of the ten segments
AG = AF = BH = BG = CK = CH = DL = DK = EF = EL = τ.
2. All five sides of the regular pentagon ABCDE are equal to τ2 , that is,
AB = BC = CD = DE = EA = τ2 .
3. All line segments AH = GC = AL = DF = EK = LC = τ2 .
4. All five diagonals of the regular pentagon ABCDE are equal to τ3 ,
that is, AC = AD = BE = BD = EC = τ3 .
5. The lengths of the segments AC, AH, HC, and GH are in geometric
progression because
AC = τ3 , AH = τ2 , HC = τ, GH = τ0 = 1.
Some interesting information about the pentagram (pentacle) is presented
in Dan Brown’s book The Da Vinci Code. The fivepointed star is clearly a pre
Christian symbol connected with the worship and dedication of nature. The
ancient people divided the world into two halves, male and female. They had
gods and goddesses maintaining the balance of forces. When a male’s and fe
male’s beginnings are balanced, then harmony is reigning in the world. When
the balance is broken, chaos appears. The pentacle symbolizes the female half of
the Universe. Historians, who study religions, name this symbol “the sacred
female beginning,” or “the Sacred Goddess.” The fivepointed star symbolizes
Venus, the Goddess of love and beauty. The Goddess Venus and the planet Ve
nus are the same. The Goddess takes her place in the night sky and is known
under many names: Venus, East Star, Ishtar, and Astarte, all of them symboliz
ing the powerful female beginnings connected with nature and mother earth.
Over an eight year period, the planet Venus describes a pentacle on the
big circle of the heavenly sphere. Ancient people noticed this unique phenom
enon and they were so impressed, that Venus and its pentacle became symbols
for perfection and beauty. Today, only a few people know that the Olympic
Games follow the half cycle of Venus; fewer people know that the fivepointed
star missed out on becoming a symbol for the Olympic Games, because of a
last moment modification: the five acute ends of the star were replaced with
the five rings. Ironically, the organizers apparently believed that their chosen
symbol better reflected the spirit and harmony of the Olympic Games.
1.7.3. The Pentagon
The Pentagon (Fig. 1.16) is the headquarters of the United States De
partment of Defense and is located in Arlington, Virginia. It is now consid
ered to be the symbol of the U.S. military. The Pentagon is often used met
Alexey Stakhov
THE MATHEMATICS OF HARMONY
30
Figure 1.16. The Pentagon
onymically to refer to the Department of De
fense — rather than the building itself. The
Pentagon is the largestcapacity office build
ing in the world, and it is one of the world’s
largest buildings in terms of floor area. It
houses approximately 23,000 military and ci
vilian employees plus approximately an addi
tional 3,000 nondefense support personnel. It
has five sides, five floors above ground (plus
two basement levels), and five ring corridors
per floor with a total of 17.5 miles (28 km) of
corridors.
1.7.4. The “Golden Cup” and the Golden Isosceles Triangle
A
The regular pentagon in Fig. 1.15 includes a num
ber of remarkable figures used widely in works of art.
In ancient art, the law of the golden cup (Fig. 1.17)
B
was widely known. It was used by the antiquity’s E
sculptors and masters of golden things. The shaded
part of the pentagram in Fig. 1.17 gives a schematic
F
representation of the golden cup.
The regular pentagon in Fig. 1.15 consists of
C
D
five golden isosceles triangles. Each golden tri Figure 1.17. The golden cup
A
angle has an acute angle
A = 36° at the top and two acute angles B = D = 72°
at the base of the triangle (Fig. 17). The main fea
36°
ture of the golden triangle in Fig. 1.18 consists of
the fact that the ratio of the hip АС=AD to the
base DC is equal to the golden mean.
In studying the pentagram in Fig. 1.15 and
H
the golden triangle in Fig. 1.18, the Pythagoreans
admired the fact that the bisector DH coincides
with the diagonal DB of the pentagon (Fig.
H''
72°
72° 1.15), and it divides the side АС in the golden
section at the point H. Moreover, the new
H' H''' C
D
golden triangle DHC arises. If we draw the bi
Figure 1.18. The golden
sector of the angle H to the point H', and con
isosceles triangle
tinue this process ad infinitum, we will get an
Chapter 1
The Golden Section
31
infinite sequence of the golden triangles. Furthermore, the case of the golden rectan
gle in Fig. 1.5 and the pentagon in Fig. 1.15, the infinite repetition of the same geo
metric figure (the golden triangle) after the drawing of the next bisector causes an
aesthetic feeling of rhythm and harmony.
1.7.5. Pentagonal Symmetry in Nature
In nature, the forms based on pentagonal symmetry (starfishes, sea hedge
hogs, flowers) are widespread. The flowers of a water lily, a dog rose, a haw
thorn, a carnation, a pear, a cherry, an appletree, a wild strawberry, and many
other plants, consist of fivepetals. Below, in Fig. 1.19, we can see examples of
nature’s structures based on the pentagonal symmetry.
The presence of five fingers on a human’s hand and, the five bone rudi
ments on the paws of many animals are an additional confirmation of the wide
spread occurrence of pentagonal forms in the morphology of flora and the bi
ological world.
Figure 1.19. Pentagonal symmetry in nature
Alexey Stakhov
THE MATHEMATICS OF HARMONY
32
1.8. The Golden Section and the Mysteries of Egyptian Culture
1.8.1. Phenomenon of Ancient Egypt
In the early 20th century, in Saqqara (Egypt), archaeologists opened a crypt
in which the Egyptian architect Hesire was buried. The wood panels covered by
a magnificent thread were extracted from the crypt together with different
material objects. In total, there were the 11 panels in the crypt; among them,
only 5 of the panels were preserved; the remaining panels were completely de
stroyed by moisture inside the crypt. On the salvaged panels the architect Hesire
was depicted with various objects and figures that had symbolic significance at
the time (Fig. 1.20).
Figure 1.20. Hesire panels
For a long time, the purpose of the Hesire panels remained vague. At first,
the Egyptologists considered these panels to be merely false doors. However,
in the mid20th century, the situation surrounding the panels began to clarify.
In the early sixties of the 20th century the Russian architect Shevelev ob
Chapter 1
The Golden Section
33
served that the sticks the architect holds in his hands on one of the panels relate
between themselves as 1 : 5 , that is, as the ratio of the small side of the 1:2
rectangle (“double square”) to its diagonal. Basing on this observation, another
Russian architect Shmelev made a very precise geometrical analysis of the Hesire
panels. This analysis led him to sensational discovery [36].
Having explored the Hesire panels, Igor Shmelev came to the conclusion:
“Now, after the comprehensive and justified analysis, by using the proportion
method, we have good ground to assert that the Hesire panels are the harmony
rules encoded in geometric language… So, in our hands we have the concrete
evidence, which shows us by “plain text” the highest standard of creative thought
of the Ancient Egyptian intellectuals. The artist who created the panels with
amazing accuracy, exquisite refinement and masterly ingenuity demonstrated
the rule of the Golden Section in its broadest range of variations. It gave birth
to the “GOLDEN SYMPHONY” presented by the ensemble of highly artistic
works, which testifies not only the ingenious talents of their creator, but also
convincingly proves that the author was aware of the secret of harmony. This
genius was the “Golden Affair Craftsman” by the name Hesire.”
But who was Hesire? Ancient texts inform us that Hesire was “a Chief of
Destius and a Chief of Boot, a Chief of Doctors, a Scribe of Pharaoh, a Priest of
Gor, a Main Architect of Pharaoh, a Supreme Chief of South Tens, and a Carver.”
Analyzing the regalia of Hesire, Shmelev paid particular attention to the
fact that Hesire was the Priest of Gor. Gor was considered by the Ancient
Egyptians as the God of Harmony, and therefore, to be the Priest of Gor he
had to execute the functions of the keeper of Harmony.
As it follows from his name, Hesire was elevated to the rank of God of Ra
(the God of the Sun). Shmelev assumes that Hesire was elevated to this rank
rewarded for his “development of aesthetic … principles in the canon system
reflecting the harmonic fundamentals of the Universe … The orientation to
the harmonic principle discovered by the Ancient Egyptian civilization was
the way to the unprecedented flowering of culture; this flowering falls into
the period of Zoser, pharaoh when the system of written signs was complete
ly implemented. Therefore it is possible to assume that Zoser’s pyramid be
came the first experimental pyramid, which was followed by construction of
the unified complex of the Great pyramids in Giza according to the program
designed under Hesire’s supervision” [36].
Thus, Shmelev states: “It is only necessary to recognize that the Ancient
Egypt civilization is the supercivilization explored by us extremely superfi
cially and it demands a qualitatively new approach to the development of its
richest heritage… The outcomes of researches of Hesire’s panels demonstrate
Alexey Stakhov
THE MATHEMATICS OF HARMONY
34
that the sources of modern science and culture are in boundless layers of history
feeding creativity of the craftsmen of our days by great ideas, which for long time
inspired the aspirations of the outstanding representatives of the mankind. And
our purpose is to keep a unity of a connecting thread” [36].
1.8.2. The Mysteries of the Egyptian Pyramids
This infinite neverending uniform sea of sand, with infrequent dried bushes,
and the barely visible tracks from a camel covered by sand, is a wasteland under
an incandescent sun. Everything around seems dull and covered by fine sand.
Suddenly, as if beset by a mirage, to our amazement we see the pyramids
(Fig. 1.21) whose fancy stone figures are directed toward the Sun. They as
tonish our imagination by their vast size and perfect geometric forms.
A Pharaoh’s authority in Ancient Egypt was tremendous; divine honors
were given to him, and a Pharaoh was called the “Great God.” The GodPha
raoh was a progenitor of the country and a judge of people’s fate. The cult of
the dead pharaoh had great significance in the Egyptian religion. The monu
mental pyramids (in addition to their more profound philosophical and math
ematical significance) were constructed for the preservation of a pharaoh’s
body, spirit, and for extolling his authority. Constructed by human hands, they
deserve to be included amongst the Seven Wonders of the World.
The pyramids had multiple functions. They served not only as vaults of a
Pharaoh’s mortal remains, but they also were the attributes of his majesty,
power, riches of country, the monuments of culture, the country’s history, and
items of information about the pharaoh, his people, and life.
It is clear that the pyramids held deep “scientific knowledge” embodied
in their forms, sizes, and orientation of terrain. Each part of a pyramid and
each element of its form were selected carefully to demonstrate the high
level of knowledge of the pyramid creators. They were constructed for mil
lennia, “for all time,” and for this reason,
the Arabian proverb says: “All in the
World are afraid of a Time. However, a
Time is afraid of the Pyramids.”
Among the gigantic Egyptian pyramids,
the Great Pyramid of the Pharaoh Cheops
(Khufu) is of special interest (Fig. 1.21).
Before the beginning of the analysis
of the form and dimensions of Cheops’
Fig. 1.21. Cheops’ Pyramid in Giza
Pyramid it is necessary to introduce the
Chapter 1
The Golden Section
35
Egyptian measurement system. The ancient Egyptians used three units of mea
surement: the “elbow” (466 mm) which equaled 7 “palms” (66.5 mm), which, in
turn, was equal to 4 “fingers” (16.6 mm). Let us analyze the dimensions of Cheops’
Pyramid (Fig. 1.22) employing considerations given in the remarkable book of
Ukrainian scientist Nikolay Vasutinsky [31].
The majority of researchers believe that the length of the side of a pyra
mid’s base is, for example, GF is equal to L = 233.16 m. This value corresponds
almost precisely to 500 “elbows.” The full accordance to the 500 “elbows” will
be, if we take the length of the “elbow” equal to 0.4663 m. The altitude of the
Pyramid (H) was estimated by researchers variously from 146.6 m up to
148.2 m and depending on the adopted pyramid’s altitude, all ratios of its
geometric construction will change considerably. What is a cause for the dif
ferences in the estimation of a pyramid’s altitude? Strictly speaking, Cheops’
pyramid is truncated. Today, its apex is approximately 10 × 10, but 1 century
ago, it was 6 × 6. Apparently, the top of the pyramid was dismantled.
In estimating a pyramid’s height, it is necessary to consider physical fac
tors, such as “shrinkage” of its construction. Under the enormous pressure
(reaching 500 tons on 1 m2 of the ground) exerted by its weight, the height of
most pyramids would have decreased in comparison to its initial designed val
ue. What was the initial height of the Pyramid? This height can be reconstruct
ed if we can discover the main or key “geometrical idea” of the Pyramid.
In 1837, English Colonel Howard Vyse measured the inclination angle of
the Pyramid faces: it appeared to be α = 51°51'. A majority of researchers
recognize this value today. The indicated value of the inclination angle corre
sponds to the tangent equal to 1.27306. This value corresponds to the ratio of
the pyramid’s altitude AC, which is half of its base СВ (Fig. 2.22), that is,
A
AC / CB = H / ( L / 2) = 2H / L .
Researchers were in a big sur
prise! If we consider the square
H
root of the golden mean, that is,
E
τ , we obtain the following out
come τ = 1.272. Comparing
this value with the value of
51°50'
51°50'
D
B
tgα = 1.27306, we see that
C
these values are very close.
If we accept the angle
α = 51°50', that is, we de G
F
crease it by one arc minute,
Figure 1.22. A geometric model of Cheops’ pyramid
the value of tgα will become
Alexey Stakhov
THE MATHEMATICS OF HARMONY
36
equal to 1.272, that is, it will be exactly equal to the value of τ . It is necessary to
note that in 1840, Colonel Vyse repeated his measurements and corrected the value
of the angle to α = 51°50'. These measurements led researchers to the following
hypothesis: the ratio AC/CB = 1.272 underlies the triangle АСВ of Cheops’ Pyra
mid! This ratio is characteristic for the above golden right triangle (Fig. 1.11).
Then, if we accept the hypothesis that the golden right triangle is the main geo
metrical idea of Cheops’ Pyramid, it is then possible to calculate the initial height
of Cheops’ Pyramid. It is equal to H = ( L / 2) × τ = 148.28 m .
Now, we deduce other relationships for Cheops’ Pyramid following from this
golden hypothesis. In particular, let us find the ratio of the external area of the
Pyramid to its base. For this purpose, we accept the length of the leg СВ for the
unit, that is, CB = 1, however, the length of the side of the Pyramid base is GF = 2,
and the area of the base EFGH will be equal to SEFGH = 4. Now, let us calculate the
area of the lateral side of Cheops Pyramid. As the altitude AB of the triangle AEF
is equal to τ, then the area of each lateral side will be equal to SD = τ , and the
common area of all four lateral sides of the pyramid will be equal to 4τ, while the
ratio of the external area of the pyramid to its base will be equal to the golden
mean. This is the main geometrical secret of Cheops Pyramid!
Analyses of other pyramids confirm that the Egyptians always intended to
embody their most significant mathematical knowledge in the pyramids they
built. In this respect, the Pyramid of Chephren (Khafre) is also rather interest
ing. The measurements of Chephren’s Pyramid show that the inclination angle
of the lateral sides is equal to 53°12', which corresponds to leg’s ratio of the
right triangle: 3:4. This leg’s ratio corresponds to the wellknown right triangle
with the side ratio: 3:4:5, this triangle is called the “perfect,” “sacred,” or “Egyp
tian” triangle. According to testimony from historians, the “Egyptian” triangle
had a magical sense. Plutarch wrote that the Egyptians compared the universe
to the “sacred” triangle; they symbolically ascribed the vertical leg to the hus
band, the base to the wife, and the hypotenuse to the child born from them.
According to the Pythagorean Theorem for the triangle 3:4:5 we have:
2
3 +42 = 52. Possibly, this famous theorem was perpetuated in Chephren’s pyr
amid based on the 3:4:5 triangle. It is difficult to find a more appropriate ex
ample to demonstrate The Pythagorean Theorem — which evidently was
wellknown to the Egyptians long before its rediscovery by Pythagoras.
Thus, the ingenious architects of the Egyptian Pyramids sought to astonish
future generations and fulfilled this goal by selecting the golden right triangle
as the “main geometrical idea” of Cheops’ Pyramid, and the “sacred” or “Egyp
tian” right triangle as the “main geometrical idea” of Chephren’s Pyramid.
Chapter 1
The Golden Section
37
1.9. The Golden Section in Greek Culture
1.9.1. Pythagoras
Pythagoras is one of the most celebrated people in the history of science. His
name is known to each person who has studied geometry and been introduced to
the Pythagorean Theorem. The famous philosopher, scientist, religious and ethical
reformer, influential politician, “demigod” — in the eyes of his followers, Phythag
oras personifies ancient wisdom. Coins with his image, which were made in 430
420 B.C., testify to the exclusive popularity of Pythagoras. In the 5th century
B.C., this was unprecedented! Pythagoras was the first, among the Greek philoso
phers, to have a special book dedicated to him.
His scientific school is internationally known. He
had organized it in Croton, a Greek colony in northern
Italy. The “Pythagorean School,” or “Pythagorean
Union,” was simultaneously a philosophical school, a
political party, and a religious brotherhood. Entrance
to the “Pythagorean Union” was very demanding. Each,
who entered the “Pythagorean Union,” would be re
fused personal property for the benefit of the Union;
they undertook to not spill blood, not eat meat, and
Pythagoras
protect the secret of their doctrine. The members of
(Born c.569 B.C.,
died c.475 B.C.)
the Union were prohibited to train others for money.
The Pythagorean doctrine focused upon harmony, geometry, number the
ory, astronomy, and other topics, but the Pythagoreans, most of all, appreciat
ed the results that were obtained in the theory of harmony because it con
firmed their idea that “numbers determine everything.” Some ancient scien
tists assume that the concept of the golden section was borrowed by Pythag
oras from the Babylonians.
Many great mathematical discoveries were attributed to Pythagoras —
some perhaps undeservedly. For example, the famous geometric “theorem of
squares” (Pythagorean Theorem), which sets a ratio between the sides of a
right triangle, was known to the Egyptians, the Babylonians, and the Chinese
long before Pythagoras.
However, discovery of “incommensurable line segments” is considered to
be the main mathematical discovery of the Pythagoreans. In investigating the
ratio of a diagonal to the side of a given square, the Pythagoreans proved that
Alexey Stakhov
THE MATHEMATICS OF HARMONY
38
its ratio cannot be expressed as the ratio of two natural numbers, that is, this ratio
is “irrational.” This discovery caused the first crisis in the history of mathematics,
as the Pythagorean doctrine about integervalued basis of all existing things could
no longer be accepted as true. Therefore, the Pythagoreans attempted to keep the
discovery a secret; they created a legend about the death of one of the Pythagoreans
— who divulged the secret of this discovery.
Also attributed to Pythagoras were a number of the other geometrical dis
coveries, namely, the theorem about the sum of the interior angles of a triangle,
and the problem about segmentation of the plane into regular polygons (trian
gles, squares and hexagons). There is a view that Pythagoras discovered some or
all of the primary regular “spatial” figures, that is, the five regular polyhedrons.
Why was Pythagoras so popular already during his life? The answer to
this question follows from interesting facts from his biography. According to
legend, Pythagoras went to Egypt and lived there for 22 years to learn from
the wisdom of the priestscientists. After studying all Egyptian sciences, in
cluding mathematics, he moved to Babylon, where he lived for 12 years and
was introduced to the scientific knowledge of Babylonian priests. Legends
also attribute Pythagoras with a visit to India. It is possible because Ionia and
Indiaat that timehad business relations. He returned home about 530 B.C.
Pythagoras attempted to organize his philosophical school. However, due to
undetermined reasons, he abandoned Samos and settled in Crotona Greek
colony in North Italy. Here, Pythagoras organized his school.
Thus, the outstanding role of Pythagoras in the development of Greek
science consists in fulfillment of the historical mission of knowledge transmis
sion from the Egyptian and Babylonian priests to the culture of ancient Greece.
Thanks to Pythagoras, who was, without any doubt, one of the leading think
ers of his time, the Greek science could obtain new knowledge in the fields of
philosophy, mathematics, and natural sciences. This new scientific knowledge
contributed to the rapid development and augmentation of Greek science in
ancient Greek culture.
In developing the idea about the role of Pythagoras in the historical de
velopment of the Greek science, Igor Shmelev, in the brochure Phenomenon
of Ancient Egypt [36], wrote:
“His world renowned name the Croton teacher obtained after the rite of
“consecration.” This name is compounded of two halves and it means “The
Prophet of Harmony” because the “Pythians” in Ancient Greece were pa
gan priests who predicted a future and Gor in Ancient Egypt personified
harmony. So in the last years of their civilization, the Egyptian priests trans
mitted their secret knowledge to the representatives of the new civiliza
Chapter 1
The Golden Section
39
tion, symbolically cementing in one person the Union of man’s and woman’s
origins, the bastion of Harmony.”
It is known that the Pentagram has always attracted special attention from
the Pythagoreans; they considered it the symbol of health. There is a legend
surrounding the pentagram: when a guest in a foreign land, one of the Pythagore
ans lay on his deathbed, unable to pay the man helping him. He asked the man
to draw the pentagram on his dwelling, hoping that this symbol would be seen
by one of the Pythagoreans. Some years later, one of the Pythagoreans saw this
symbol and the kind host received a rich compensation.
Already, the fact that the Pythagoreans chose the “pentagram,” brimming
with golden sections, as the main symbol of their Fraternity, is another confir
mation that the Pythagoreans knew and esteemed the golden section. The
Russian researcher Alexander Voloshinov wrote [137]:
“The Pythagoreans gave special attention to the pentagram, the fivepoint
ed star that is formed by diagonals of the regular pentagon. In the pentagram,
the Pythagoreans found all proportions wellknown in antiquity: arithmetic,
geometric, harmonic, and also the wellknown golden proportion, or the golden
ratio. A perfection of mathematical forms of the pentagram finds a reflection in
the perfection of its form itself. The pentagram is proportional and, hence, beau
tiful. Probably owing to the perfect form and the wealth of mathematical forms,
the pentagram was chosen by the Pythagoreans as their secret symbol and a
symbol of health. Thanks to the Pythagoreans, the fivepointed star is today a
symbol for many states and is on the flags of many countries of the world.”
1.9.2. The Idea of Harmony in Greek Culture
The idea of harmony based on the golden mean became one of the most
fruitful ideas in Greek art. Nature, taken in a broad sense, includes the human
creative patterns of art, where the same laws of rhythm and harmony domi
nate. Aristotle states:
“Nature aims for the contrasts and from them, instead of similar things,
Nature forms a consonance …. This combines a male with a female and thus
the first public connection is formed through the connection of contrasts,
instead of similarities. Also in art, apparently, by imitating Nature the artist
acts in the same way. Namely the painter makes the pictures conform to the
originals of Nature by mixing white, black, yellow and red paints. Music
creates the unique harmony by mixing different voices, high and low, long and
short, in congregational singing. Grammar creates its entire art from the
mixture of vowels and consonants.”
Alexey Stakhov
THE MATHEMATICS OF HARMONY
40
Major
Minor
To take the subject and eliminate all that is superfluous is the plan of the
ancient Greek artist. This is the main idea of Greek art, where the golden
section lay at the center of the canon of aesthetics. The theory of proportion
is the foundation of Greek art, and, the problem of proportionality could not
have escaped the thought of Pythagoras. Among the Greek philosophers,
Pythagoras was the first who attempted mathematically to understand the
essence of musical or harmonic ratios or intervals. Pythagoras knew that the
intervals of the octave can be expressed by natural numbers, which fit the
corresponding oscillations of the string, and these numerical relations were
put forth by Pythagoras as the basis of his theory of musical harmony. Pythag
oras was credited with possessing knowledge of arithmetic, geometric, har
monic proportions, and the law of the golden section. Pythagoras paid special
attention to the golden section by choosing the pentagram as the distinctive
symbol of the “Pythagorean Union.” By developing further the Pythagorean
doctrine about harmony, Plato analyzed the five regular polyhedrons (the so
called Platonic solids) and emphasized their ideal beauty.
As to the main requirements of beauty, Aristotle proposed order, proportion
ality, and size limitation. The order arises when certain relations and proportions
hold between the whole and its parts. In music, Aristotle recognized the octave as
the most beautiful consonance; he considered that the ratio of oscillations be
tween the basic tone and its octave is expressed by the first natural numbers: 1:2.
He also maintained that in poetry, the rhythmic relations of a verse are based on
small numerical relations, and by this, a beautiful impression is expressed. Except
for the simplicity based upon the commensurability of separate parts and the
whole, Aristotle, as well as his mentor Plato, recognized
that the highest beauty of perfect figures and propor
tions was based upon the golden section.
The ancient Harmony theory, based on the prin
ciple of the division in extreme and mean ratio (the
golden section), became the “launching pad,” upon
which, subsequently, a harmony concept was devel
oped in the science and art of European culture.
1.9.3. The Golden Section in Greek Sculpture
For a long time, the creations by the great Greek
sculptors Phidias, Polyclitus, Myron, and Praxiteles
were considered to be the standard of beauty and har
monious construction based on the human body. The
Figure 1.23. Doryphorus
by Polyclitus
Chapter 1
The Golden Section
41
0,618
0,382
statue of Doryphorus, created by Polyclitus in the 5th century B.C., serves as
one of the greatest achievements of classical Greek Art (Fig. 1.23). This statue
is considered to be the best example for analysis of the proportions of the ideal
human body established by ancient Greek sculptors. It is especially important
because the name “Canon” was ascribed to this famous sculpture. The harmon
ic analysis of the Doryphorus as given in the book Proportionality in Architec
ture (1933) [10], written by the Russian architect G.D. Grimm, indicates the
following connections of the famous statue to the golden mean :
1. The first golden cut division of the Doryphorus figure with its overall
height M 0=1, relating segments M 1 and M 2, is at the navel.
2. The second division of the lower part of the torso, relating M 2 and M 3,
passes through the line of his knee.
3. The third division, relating M 3 and M 4, passes through the line of his neck.
The Venus de Milo (Fig. 1.24), a statue of the goddess
Aphrodite, is one of the bestknown monuments of Greek
sculptural art. This statue was created by Agesandr in the
2nd century B.C. The Goddess Aphrodite is represented
halfnaked, so that her clothing — which wrap up her legs
and the bottom of her torso, as if it were a pedestal for the
open hands, which are showing in movement (Fig. 1.24).
During the Hellenic epoch, Aphrodite was one of the most
favorite goddesses. From the Island Melos, Aphrodite is
strict and restrained. It is thought that she stood on the
high pedestal and looked at over the spectators. The Ve
nus de Milo is a pearl of the Louvre; it is a standard of
Figure 1.24. Venus de Milo female beauty in ancient Greece.
1.10. The Golden Section in Renaissance Art
1.10.1. The Idea of the “Divine Harmony” in the Renaissance Epoch
The idea of Harmony belonged to such ancient conceptual ideas, which
attributed to the interest of the church. According to Christian doctrine,
the universe was created by God and submitted to its will unconditionally.
The Christian God, at the creation of the universe, was guided by mathe
Alexey Stakhov
THE MATHEMATICS OF HARMONY
42
matical principles. In Renaissance culture, Catholic doctrine included the math
ematical plan to which God created the universe.
In the opinion of the American
historian of mathematics, Morris
Klein, the close merge of the reli
gious doctrine about God as the cre
ator of the universe, and the antique
idea of numerical harmony of the uni
verse, became one of the major sta
ples of the Renaissance culture [6].
Mainly, the objective of Renaissance
science is presented in the following
text from the wellknown astrono
mer Jogannes Kepler:
“The overall objective of all re
Figure 1.25. Holy Family by Michelangelo searches of the external world should
be to discover the rational order and harmony, which was embodied by the
God in the Universe and then was presented for us by the God on
mathematics language.”
The art of the Renaissance period (espe
cially paintings) is substantially connected to
topics from the Bible. The picture Holy Fam
ily by Michelangelo is a bright example of
such topical art. This picture is a fairly recog
nized masterpiece of WestEuropean art. The
picture is often named “Tondo Doni” because,
firstly, the picture belonged to the Doni fam
ily in Florence, and secondly, it had a round
form (in English “tondo”). After the harmon
ic analysis of this picture, researchers found
that a compositional construction of the pic
ture is based on the pentacle (Fig. 1.25).
The picture Crucifixion, by Rafael San
ti (14831520), is another example of a pic
ture based on the topics of the Bible. The
harmonic analysis of this picture (Fig. 1.26)
showed that the compositional plan of the
picture is based on the golden isosceles tri
Figure 1.26. Crucifixion by
Rafael Santi
angle (Fig. 1.18).
Chapter 1
The Golden Section
43
1.10.2. “Vitruvian Man” by Leonardo da Vinci
During the age of the Renaissance, attempts to create an ideal model of a
harmoniously developed human body were continuing. It is known, that a sweep
of human hands approximately is equal to a
human growth. It means that the human fig
ure can be inserted into a square and a circle.
The ideal human figures, created by Leonar
do da Vinci and Durer, are widely known. For
a long time, the opinion existed that the “pen
tagonal” or “fivefold” symmetry, which is
characteristic for flora and animals, is shown
in the structure of the human body. The hu
man body can be considered as a sample of
the pentagram, where the human head, two
hands, and two legs are, as if, beams of the
pentagonal star. Such model found a reflec
tion in the constructions of Leonardo da Vin
ci and Durer, in particular, in the figure of the
wellknown Vitruvian Man by Leonardo da
Figure 1.27. Vitruvian Man
Vinci (Fig. 1.27).
by Leonardo da Vinci
1.10.3. “Mona Lisa” by Leonardo da Vinci
The Renaissance, in the cultural history of the Western and Central
European countries, is a transitive epoch, whereby it progressed from
the Medieval culture to the culture of a new era. A humanistic world
outlook and a return toward the antique cultural heritage, as though, “Re
naissance” of the ancient culture, are the most typical features of this
epoch. The Renaissance is characterized by large scientific shifts in the
field of natural sciences. The close connection to art is a specific feature of
science of this epoch, and this feature was sometimes expressed by one
creative person. Leonardo da Vinci, the outstanding artist, scientist, and
engineer of the Renaissance, was a brilliant example of a manysided per
son. By his nature, Leonardo da Vinci had enviable health, was goodlook
ing man, tall, and blueeyed. Leonardo was born on the 15th of April under
the Star of Mars, and possibly, therefore, he possessed a huge force and
man’s valour. He sang marvelously by composing melodies and verses for
his listeners. He played many different musical instruments; moreover, he
created new musical instruments.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
44
But art and science were the main spheres of Leonardo’s
creative work — where Leonardo showed his genius talent.
For Leonardo’s art works, his contemporaries and descen
dants gave such definitions, as “genius,” “divine,” “great,”
but the same words can be used to describe Leonardo’s sci
entific discoveries. He invented a military tank, a helicopter,
an underwater ship, a parachute, an automatic weapon, a
diving helmet, an elevator, and so on. He solved the most Leonardo da Vinci
complicated problems of acoustics, botany, and cosmology; he invented an hour
pendulum one century earlier than Galileo, and he developed a mechanics theory
— among other things. In his relationship with Universe Harmony, Leonardo
expressed the following words: “All earth, mountains, woods, and seas form a
whole, in which each thing feeds the other one, all things are cooperated and
interconnected, supported, but at the same time destroyed and updated.”
Seemingly, each visitor of the Louvre in Paris tries to find the famous “Mona
Lisa” (“Jokonda”) by Leonardo da Vinci. The Great Artist drew this portrait tense
ly and long. He made many sketches; he paid attention to the pose of Mona Lisa, to
the turn of her head, and to the position of her hands. Italian artist Giorgio Vazari
(15111574) tells in his “Biographies,” that during the painting of Mona Lisa, Le
onardo invited singers, musicians, and clowns to his studio to support the cheerful
mood of the young woman and to have an opportunity to watch a changeable
expression of her face. And only after four years of intense work he, at last, could
present to cultural community his worldfamous “Mona Lisa” (“Jokonda”).
It is considered, that a secret of Jokonda’s charm is in the variability of her
smile. There is an opinion that the woman represented in the picture had lost a
child, several months ago. The dark color of her clothes speaks about this. She sits
in a quiet pose combining her hands in her lap, but her face is full of imperceptible
movement: her lips tremble in light smile, and her smiling eyes attract spectators
attentively and derisively. In those days, the lips, slightly opened in corners of the
mouth, were considered as an attribute of elegance. Jokonda’s hardly appreciable
smile, gentle and mysterious, lights up the picture. Mysteriousness of the image
is strengthened by the background, a mountainous silveryblue landscape.
What is the reason for Jokonda’s charm? The search for the answer to this
question continues. By creating the masterpiece, Leonardo used a secret known
for many portraitists: the vertical axis of the picture passes through the center of
the left eye what should cause the feeling of excitation of the spectator, that is, in
the picture the artist used a “Principle of Symmetry,” But, possibly, the reason is
in another. The picture of the genius artist attracted attention of the researchers.
After precise analysis, they found two golden triangles one with its base stand
Chapter 1
The Golden Section
45
ing on the bottom of the picture, another with its
base touching the top of the picture and the top
touching the bottom of the picture, and their com
mon height crosses the center of Jokonda’s left
eye. (Fig. 1.28). Further harmonious analysis of
the picture showed, that the center of Jokonda’s
left eye is on the crossing of the two bisectors of
the upper golden triangle, which, on the one hand,
bisect the angles at the base of the upper golden
triangle, and, on the other hand it divides the sides
of this golden triangle in the golden section. Thus,
Leonardo used in the picture not only the “Prin
ciple of Symmetry,” but also the “Golden Section
Principle.”
What is considered the apex of Leonardo’s
Figure 1.28. Harmonic Analysis creativity, this picture is thought as the crys
of Mona Lisa (“Jokonda”)
tallization of his genius, innermost thoughts,
by Leonardo da Vinci
and inspiration. Very little is known about
Mona Lisa, except for several insignificant facts, therefore, it is difficult to
answer very important questions often asked and discussed: whether she was
simply a beautiful model for Leonardo or she was his muse and even his love.
There are some facts, which confirm the last assumption, and this, probably,
explains a special magic of the picture. But it is clear that Mona Lisa was that
woman who inspired Leonardo to create this unique masterpiece inspiring
thousands of people over many centuries.
A huge number of legends are connected with this wellknown picture. We
will begin from the history of its creation. Giorgio Vazari, in his “Biographies,”
wrote the following: “Leonardo undertook to execute for Franchesco Del Jokon
do the portrait of his wife Mona Lisa.” Some researches assume that Vasari,
apparently, was mistaken. The newest investigations showed that in the pic
ture another woman, not Mona Lisa, the wife of the Florentine nobleman Del
Jokondo, was presented. Many researchers speculate why Del Jokondo refused
the portrait of his wife, but in reality the portrait became the property of the
artist, and this fact is an additional argument favoring the fact that Leonardo
represented another woman and not Mona Lisa. However, there is another legend
why Mona Lisa’s portrait was found at Leonardo — simply Leonardo had drawn
two copies of this famous picture.
Probably, the creation of this picture is connected with some secret in
Leonardo’s life. Leonardo’s riddle begins with his birth. As is known, Leonar
Alexey Stakhov
THE MATHEMATICS OF HARMONY
46
do was an illegally born son of a woman, about which almost nothing is known. It
is known only that her name was Katerina and that she was the owner of a tavern.
Not much more information is known about Leonardo’s father. Mr. Pero,
Leonardo’s father, was 25 years old when Leonardo was born. He was the notary,
and he possessed impressing male merits: he had lived till the age of 77s, had four
wives (three of them he buried), fathered 12 children and the last child was born
when he was 75 years old.
In the Renaissance, a relationship with illegitimate children was tolerat
ed. Leonardo, at once, was recognized by the family of his father. However, in
the house of his father he was taken away after awhile. Soon after his birth,
Leonardo was sent together with Katerina to the village Anhiano located near
to the city of Vinci. He remained there about four years. During these years,
Mr. Pero married his first wife — a 16yearold girl — who had a higher social
position than Leonardo’s mother.
The young wife had turned out fruitless. Probably, for this reason, Le
onardo at the age of almost 5 was taken in his father’s city house, where at
once he was cared by his numerous relatives: grandfather, grandmother, fa
ther, uncle, and foster mother.
During his life, Leonardo always remembered his native mother Katerine
who surprisingly resembled Mona Lisa, the wife of the Florentine merchant
Jokondo, and, probably, this fact became the main reason of Leonardo’s desire
to create Mona Lisa’s portrait. He embodied into this picture everything that
was cheerful, light, and clear to him. He embodied all of a son’s love to his
mother Katerine in this wellknown picture, which defined the development
of painting for many centuries forward, and we should thank God that Le
onardo da Vinci met Mona Lisa during the final stage of his creative way.
1.11. De Divina Proportione by Luca Pacioli
1.11.1. Luca Pacioli
The spiritual heritage of Greece, Rome, and Byzantium were marvelous
ly combined in the new Renaissance lead by its Titans of gigantic intellect
and artistic talent. “Titan” is the most appropriate term for the likes of Le
onardo da Vinci, Michelangelo, Nicholas Copernicus, Albrecht Durer, Raphael,
Bramante and many others. Luca Pacioli, the Italian mathematician of the
Chapter 1
The Golden Section
47
Renaissance, collaborated with Leonardo da Vinci, was the sem
inal contributor to the field now known as accounting, takes a
place of pride amongst the Titans. He was also called Luca di
Borgo after his birthplace, Borgo Santo Sepolcro, Tuscany,
where he was born in 1445.
Luca Pacioli is rightfully called “The Father of Account
ing” and “The Unsung Hero of the Renaissance.” In fact,
the Fransican Friar Luca Pacioli was one of the most re
markable people of his epoch, but unfortunately was one of Luca Pacioli
(1445 1514)
the least wellknown. This is surprising because his work as
a master of math is brilliant and revolutionary and continues to affect us.
Although we know little of Pacioli’s early life, there is evidence that he
began to study in the art studio of the artist Piero della Francesca. Amongst
his contemporaries, Pierro della Francesca was known as a mathematician
and geometer, as well as, an artist. Today he is chiefly appreciated for his art.
Through Piero, Pacioli gained access to the Count of Urbino and the library
of Federico where he gained access to thousands of books. This allowed Luca
to expand and deepen his knowledge of mathematics. Piero also would later
introduce Pacioli to his new mentor, Leon Battista Alberti, who became the
second great man in Luca Pacioli’s life. The following words of Alberti fell
deeply into Luca’s consciousness: “We may conclude Beauty to be such a
Consent and Agreement of the Parts with the Whole in which it is found, as to
Number, Finishing and Collocation, as Congruity, that is to say, the principal
Law of Nature requires. This is what Architecture chiefly aims at, and by this
she obtains her Beauty, Dignity and Value.”
Alberti introduced Luca to Pope Paul II. Pope Paul encouraged Luca to
dedicate his life to God and become a monk. When Alberti died in 1472, Luca
took the vows of a Franciscan Minor.
In 1475 Pacioli began to work at the University of Perugia, receiving a
professorial chair in 1477. He remained at the University for six years while
becoming the first lecturer holding a chair in math in this University. While
there Pacioli wrote a mathematical manuscript dedicated to the “Youth of
Perugia.” After he left Perugia, he took up more traveling and wandered
throughout Italy, but in 1486 was called back to the University of Perugia by
the Franciscans. It was at this time that he started calling himself “Magister,”
or “Master” — the equivalent of what we today refer to as a full professorship.
Luca wrote his famous work Summa de Arithmetica, Geometria, Propor
tioni et Proportionalita, when he was 49 years of age. Luca did this out of the
belief that mathematics was being poorly taught at this time.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
48
One section of the book that was entitled Particularis de Computis et Scrip
turis was dedicated to accounting. This book was referred to by some as “a
catalyst that launched the past into the future.” Luca was the first person to
explain the doubleentry system, or the Venetian method. This was absolute
ly revolutionary and far ahead of its time, sealing for him the title “The Father
of Accounting.” The Summa was recognized for many accomplishments and
it became the most widely read mathematical work in all of Italy.
After Leonardo Da Vinci read Pacioli’s Summa, he arranged for Luca to
come to the Court of Duke Lodovico Maria Sforzo to tutor him in mathemat
ical perspective and proportion. Luca joined Leonardo at the Sforza Court in
1496, whence began a sevenyear relationship that produced two enduring
masterpieces. Under the direct influence of Leonardo da Vinci, Luca Pacioli
began to write the book De Divina Proportione, pub
lished in 1509. The book had a noticeable influence on
his contemporaries and was one of the first fine exam
ples of the Italian art of bookprinting. The historical
significance of this book is that it was the first mathe
matical book solely dedicated to the golden mean. Da
Vinci used his artistic skills to illustrate Luca’s De Div
ina Proportione, the second important Pacioli manu
script. At the same time, Luca taught Leonardo per
spective and proportionality. This knowledge would
remain with him forever, and help him to create one of
his greatest masterpieces, “The Last Supper.” Painted
on the back wall of the dining hall at the Dominican
Figure 1.29.
convent of Santa Maria delle Grazie in Italy, it instant
Luca Pacioli’s book
ly became one of the most famous works of the Italian
De Divina Proportione
Renaissance (if not of all time).
De Divina Proportione consists of three parts: the first part sets out the
properties of the golden ratio, the second part is dedicated to the five regu
lar polyhedrons, and the third part describes the applications of the Golden
Section in architecture. Using Plato’s Republic, Timeaus, and Laws Pacioli
sequentially infers the twelve different properties of the Golden Section.
Pacioli characterizes them using such epithets as “exceptional,” “remark
able,” “almost supernatural,” etc. Considering his view (and that of the an
cients) that the Golden Section is the universal ratio that expresses a per
fection of Beauty in Nature and Art, he names it the Divine Proportion and
recognizes it as a “thinking tool”, the “aesthetic canon”, and the “Funda
mental Principle of the Cosmos”.
Chapter 1
The Golden Section
49
This book is one of the first mathematical works that attempted to provide a
solid scientific basis fot the Christian doctrine about God, the creator of the
Cosmos. Considering the properties of the golden mean inherent in God, Pacioli
names it the “Divine Proportion.”
1.11.2. About Pacioli’s Plagiarism
Roger HerzFischler wrote in his book [40]: “Perhaps no mathematician has
plagiarized as much and with so little change in details and has stirred so much
controversy and such vehement reactions as Luca Pacioli.” The main charge for
plagiarism falls on Pacioli’s De Divina Proportione. Some have claimed that this
book is a literal translation out of the Latin of Piero della Francesca’s manuscript
De Quinque Corporibis. However, De Divina Proportione is not the only book of
Pacioli that falls under this charge. According to Davis [40], Pacioli also derived
dodecahedron and icosahedron problems from Piero’s Trattato and included them
in his 1494 book, Summa de Arithmetica, Geometria, Proportioni et
Proportionalita.
It is now surprising why Pacioli’s books caused such vehement reaction.
Euclid, for instance, could also be blamed for plagiarism in The Elements. Ac
cording to the statement of the famous mathematician and historian Van der
Waerden, a majority of the mathematical results stated in The Elements be
longed to the Pythagoreans. The following quote from Roger HerzFischler’s
book [40] is very interesting in regard to the aforementioned:
“As a final comment on Pacioli’s “literary borrowings,” I mention Cardano’s
16th century comments … on Pacioli’s use in his Summa of a 1202 manuscript
(Fibonacci?) dealing with algebra; and Agostini (1925) who, after discussing
Pacioli’s word for word inclusion in his Summa of a 1481 mercantile text, says
that one should not accuse Pacioli of plagiarism for having used in his book
what is available to everybody and is not personal intellectual property.”
Thus, following the logic of Pacioli’s accusers, all modern books on geom
etry are plagiarized, as they could not be written without using Euclid’s Ele
ments! Unlike artistic creative work, a distinctive feature of scientific, in par
ticular, mathematical creative work, is that ideas of preceding fundamental
scientific results can be used by all subsequent authors in new articles and
books. Therefore, it is our responsibility as scientists, who are often in the
same position, to stand up for the great Italian mathematician, Luca Pacioli,
against such accusations of plagiarism. As mentioned earlier, Piero della
Francesca was the first teacher and very good friend of Luca Pacioli. It is
more Francesca’s mathematical works than his art works that influenced the
Alexey Stakhov
THE MATHEMATICS OF HARMONY
50
young Luca. There is absolutely nothing extraordinary in Luca’s use of Piero
della Francesca’s mathematical ideas. At the very moment Pacioli began
working on his book, De Divina Proportione; he was already a wellknown
mathematician and the author of the book Summa de Arithmetica, Geometria,
Proportioni et Proportionalita. The latter could well be considered to be the
mathematical encyclopedia of the Renaissance. Considering these facts, it is
hard to believe that the book De Divina Proportione is literal translation of
Francesca’s manuscript De Quinque Corporibis. More likely, Luca Pacioli used
it as the basis of his “second great book” De Divina Proportione. Leonardo da
Vinci unquestionably was the illustrator of this remarkable book with its 60
geometrical figures. Perhaps somebody will argue that he borrowed them
from Piero della Francesca’s works. Following the logic of the Pacioli accus
ers, all future books in the field of the “Golden Section,” including the present
book, should be recognized as a “plagiarism” of Pacioli’s book De Divina Pro
portione, the first book focusing on the Golden Section in history!
1.11.3. Death and Oblivion of Luca Pacioli
Already tired and ill, in 1510, Luca Pacioli was 65 years old. In the fore
word of his unpublished book About the Forces and Quantities he wrote the
sad phrase: “The last days of my life approach.” This manuscript is stored in
the library of the Bologna University. Pacioli died in 1515 and is buried in the
cemetery of his native town Borgo SanSepolcoro.
After Pacioli’s death, his works fell into oblivion for almost four centuries.
At the end of the 19th century, his works became internationally wellknown.
After the 370year hiatus of recognition, grateful descendants in 1878 placed
a memorial stone in Pacioli’s house of birth in Borgo San Sepolcro with the
following inscription that reflects the essential significance of Pacioli:
“For Luca Pacioli, who had da Vinci and Alberti as friends and advisers,
who turned algebra into a science, and applied it to geometry, who lectured in
doubleenrty bookkeeping, whose work was the basis and the norm for later
mathematical research; for this great fellow citizen, the people of San Sepul
cro, ashamed of their 370 years of silence, have placed this stone, in 1878.”
The names of three geniuses of the Renaissance are mentioned in this
inscription on the stone: Leonardo da Vinci, Leon Battista Alberti and Luca
Pacioli. Their works became a valuable if not priceless contribution to the
development of the theory of Harmony and the Golden Section!
Pacioli’s memorable work is not forgotten in 20th and 21st centuries.
A beautiful Monument to Luca Pacioli was sculpted out of white Carrara
Chapter 1
The Golden Section
51
marble and placed
in Sansepolcro, It
aly in honor of the
anniversary of the
1494 publication of
the Summa. In re
cognition of Pacio
li’s great works, a
500 lira post stamp
was issued, and nu
merous conferenc
es were held around
the world.
Memorial stone in Borgo San Sepolcro, 1878
In honor of 500 anniversary of Summa, 1994
1.12. A Proportional Scheme of the Golden Section in Architecture
The 1935 book Proportionality in Architecture [10] by Russian architect
Professor Grimm is well known in the theory of architecture. The purpose of
the book is set forth in its Introduction:
“In view of the exceptional significance of the golden section as the pro
portional division that establishes a constant connection between the whole
and its parts and gives a constant ratio between them, unachievable by any
other division, it is foremost the normative basis upon which we will subse
quently check the proportionality of both historical and modern monuments
and constructions….
Alexey Stakhov
THE MATHEMATICS OF HARMONY
52
Taking into consideration this general importance of the golden section in all
manifestations of architectural thought, it is necessary to recognize the impor
tance of it, as the basis of architectural proportionality theory in general.”
Grimm illustrates the golden section as a division of line segment AB at point
C into two unequal parts where the larger part CB is called major, and the smaller
part AC is called minor. The whole line AB is to the major (segment CB) as the
major (segment CB) is to the minor (segment AC), both in the golden ratio. Fol
lowing a detailed analysis of the properties of the golden section, and in a manner
not unlike Pacioli’s comparison of it to the qualities Divine, Grimm places the
golden mean in the forefront of all other proportions. He writes:
“In general it is necessary to recognize the extremely outstanding prop
erty of the golden section, which cannot be reached by arithmetic mean pro
portions and other divisions of the whole.”
Grimm further illustrates examples of linear proportionality of the gold
en division (see the Doryphorus statue, Fig. 1.23), through analysis of golden
divisions of rectangles, triangles, circles, and golden spirals. And finally, he
reviews volumetric proportionality through divisions of cubes, parallelepi
peds, triangular prisms, and tetrahedral pyramids.
“This analysis of the significance of the golden section and its exclusive
properties for the theoretical solution of proportional division problems of
linear, planar and volumetric masses, leads us to the following conclusion: For
full proportional coordination of architectural structures representing a volu
metric solution, it is necessary to have a golden proportioning, not only of its
linear dimensions in vertical and horizontal, but
also of the planar areas, and thence, of all volumes.”
Grimm validates his theoretical researches
in the field of the golden proportional scheme by
architectural examples from classical art (Par
thenon, Jupiter’s temple in Tunis), Byzantine
monuments, and Italian Renaissance art and ar
chitecture (Saint Peter’s Cathedral, Fig. 1.30;
and the Calleoni monument).
Grimm also analyzed structures of the Baroque
period as they differ from the architecture of the
Classical period and the Italian Renaissance, and at
first sight may have been lacking the golden sec
tion. He analyzed the Smolny Cathedral (Fig. 1.31)
Figure 1.30. Saint Peter’s
Basilica in Rome
and its surrounding monastery in St. Petersburg,
(architect Bramante)
Russia, a shining example of Baroque style. The
Chapter 1
The Golden Section
53
complex was commissioned in 1744 by Empress Eliza
beth, a daughter of Peter the Great, and was construct
ed in place of the Admiralty’s resin yard. Construction
commenced in 1749 under the leadership of the archi
tect Rastrelli, author of the Winter Palace, and the ba
sic structure was completed in 1764. The magnificent
bluewhite cathedral rising to a height of 93.7 meters
stands at the center of the Smolny monastery com
plex. The cathedral is stunning in its luxury, perfection
of proportions, and variety of decorative forms. It pos
sesses a strange, mystical peculiarity and magnificence
Figure 1.31. Smolny
that is strengthened as one approaches it. After ana
Cathedral St.Petersburg,
lyzing the cathedral, Grimm concluded that its archi
Russia
tecture was based upon the golden section.
Although there is no generally accepted opinion amongst architects
as to Grimm’s harmonious views, his editor’s foreword to Proportionality
in Architecture states:
“Nevertheless, his attempt at a general formulation of the golden section prin
ciple as the basis of proportionality in a variety of architectural styles, support
ed by analyses of material from ancient and European architecture, deserves to
be published. All the more, Grimm’s book presents a historical sketch of the
development and use of proportionality theory, and also a comprehensive math
ematical statement of the principle of the golden section.”
1.13. The Golden Section in the Art of the 19th and 20th Centuries
1.13.1. Ivan Shishkin’s Picture “The Ship Grove”
Ivan Shishkin (18321898) holds a place of honor in Russian painting.
The entire history of Russian landscape painting in the second half of the
19th century is associated with his name. The art works of this eminent
master took on special significance and became established as classics of
Russian painting. Shishkin’s creative work is an exclusive phenomenon
amongst the masters of painting. Like many Russian artists he possessed
a tremendous native talent.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
54
Major
Minor
Minor
Major
The picture The Ship Grove (Fig.
1.32), created by Shishkin in 1898, is his
last work. It is considered a worthy cul
mination to his original and creative ca
reer. This picture has a perfect classical
form, from the point of view of com
pleteness and composition. The natural
etudes created by Shishkin in his native
land’s forests, where he found his own
Figure 1.32. Harmonic analysis of
idealistic synthesis of harmony and
greatness, were thought to be the basis of his landscapes. In The Ship Grove the
artist exemplified a thorough knowledge of Russian nature that was infused within
him throughout his almost fifty years of creative life.
It is obvious that the golden section law can be found in this wellknown
picture, The Ship Grove (Fig. 1.32). For example, the brightly sunlit foreground
pine divides the horizontal plane of the picture in the golden section. Further
more, from the right of the pine there is the sunlit hillock. This hillock divides the
vertical plane of the picture in the golden section. From the left of the foreground
pine we can see many other pines. A division of the left part of the picture’s
horizontal plane in the golden section suggests itself. Presence of the bright ver
tical and horizontal lines that divide the picture in the golden section produces a
balance and calmness consistent with the artist’s intention.
1.13.2. “Modulor” by Le Corbusier
Le Corbusier (18871965) is an exceptional archi
tect and theorist. His buildings became a triumph of
the new architectural aesthetics with their vitality and
humanism. After his book To Architecture was published
in 1923, Le Corbusier became one of the twentieth
century’s leading architectural theorists.
During the Second World War Le Corbusier creat Figure 1.33. Le Corbusier’s
Modulor
ed his Modulor, a system of new proportional relations.
He used the proportions of the human body as a principle of architectural met
rics and, as a basis of his Modulor, he took not only the average height of a man, but
also man’s measurements while seated, including lengths of the hand and foot.
The Modulor is not just a theory; it is a practical guide for the employment of
human proportions in architecture. Einstein was one of the first great scientists
who appreciated the Modulor for its true value. He recognized that Corbusier’s
Chapter 1
The Golden Section
55
system had practical importance not only for architec
ture, but for other kinds of human activity as well. Har
monic analysis of the Modulor (Fig. 1.33) leaves no
doubt of the presence of the golden section.
1.13.3. The Building of the United Nations
Headquarters in New York
Figure 1.34. The Building
of the United Nations
Headquarters in New York
The construction of this world famous building
(Fig. 1.34) is associated with the name of Oscar Ni
emeyer (19071989), wellknown twentieth centu
ry Brazilian architect. He was the main adviser at
the construction of the United Nations headquar
ters in New York. Its main architectural idea of this
building consists of the three golden rectangles.
1.13.4. A Picture “Near to the Window” by Konstantin Vasiliev
Konstantin Vasiliev (19421976) is a modern Russian artist who unfortu
nately died quite young. He lived and worked near Kazan in the village Vasilevo.
His popularity is owed to his mixture of traditional symbolism and modernism
through the painting of folklore themes from ancient and modern Russian histo
ry. He was first introduced to the golden section at the Kazan school of art.
Having studied the compositional principles of the golden section used by an
cient Greeks, Konstantin decided “to investigate harmony through algebra.”
He began exploring the possibility of applying the laws of harmony to the
whole picture area in order to achieve maximal artistic expression of the image.
He was moving the different segments expressing the golden section within the
picture. Lastly, using the golden spiral, Vasiliev defined for himself how the hu
man eye perceives the subject of the picture. Thus he would find how the picture
should be constructed, and where he should
set the imaginary main point of the picture
so that it would absorb all the plot lines.
Vasiliev’s Near to the window (1975) is
a fine example of construction using the
golden mean in all aspects of the picture
(Fig. 1.35). We can only guess what the art
ist was trying to tell us. The main idea of Figure 1.35. Vasilev’s picture Near to
the window (1975)
this picture and all of its ramifications lies
Alexey Stakhov
THE MATHEMATICS OF HARMONY
56
in the face of the young woman glowing with cleanliness, dignity and quiet wis
dom. The artist framed her face around the golden point of the picture that is
located on the crossing of the two golden lines, horizontal and vertical. Therefore,
this compositional decision is one of the reasons for the feeling of harmony of the
picture that manifests within all primordial beginnings and the beauty of Russian
women.
1.14. A Formula of Beauty
1.14.1. The Golden Ratio in the External Forms of a Person
Countless artists, poets and sculptors have admired the beauty of the
human body! The French sculptor Rodin proclaimed, “The naked body seems
to me beautiful. For me it is a miracle where there can not be anything ugly.”
A human is considered the highest creation of Nature. Therefore, a human
body at all times is recognized as the most perfect and worthy object of sculp
tural art. The problem of correct proportional representation of the human body
was always one of the most important factors in art. The golden mean played a
leading role in the art canons of Leonardo da Vinci, Durer and other great
artists. According to these canons the golden section relates the whole body to
its two unequal subdivisions cut at the waistline. This is the most simple realiza
tion of the proportional division of the human body (Fig. 1.36a). In addition,
the golden proportion is often used by designers of clothes (Fig. 1.36b).
b)
a)
Figure 1.36. The golden mean in human
body (a) and clothes design (b)
Chapter 1
The Golden Section
57
1.14.2. The Beauty of a Woman’s Face
Repeated attempts have been made to analyze a woman’s face using the golden
mean and pentagram (Fig. 1.37).
Figure 1.37. A harmonious analysis of a woman’s face
Numerous researchers have come to the general conclusion that a woman’s
face is beautiful due to the golden section relations exemplified in it. The face of
a woman displays an array of emotions that are an integrated element of her
beauty. It is proven that a woman’s face fits the proportions of the golden section
most fully when she smiles. Any woman is perceived as more beautiful with a
warm smile than with a rigid face filled with anger, arrogance and disregard. Ad
mirers of great feminine beauty would be advised to make note of these golden
section principles of aesthetics.
1.14.3. Nefertiti
During excavations in 1912, a German archaeological expedition digging in
the deserted city of Amarna discovered a ruined house and studio complex. The
building was identified as a sculptor’s workshop. One of the items found had the
name and job title of Thutmose, court sculptor of Egyptian Pharaoh Akhenaten.
Among the many sculptures recovered was the famous head of Queen Nefertiti
(in ancient Egyptian meaning the “beauty is coming”). This sculpture is
Alexey Stakhov
THE MATHEMATICS OF HARMONY
58
recognized as a symbol of feminine beauty. After the exposition of Nefertiti in
the Berlin museum, her fame rose as if she were a modern movie star rather than
an ancient Egyptian queen.
With a charming head, long harmonic neck, and direct, but gently outlined
nose, the Nefertiti sculpture caused increased interest in Egyptian art, including
its deep mystical past of a cult of priests and esoteric wisdom. Possibly, our irra
tional century has selected Nefertiti as a symbol of unusual feminine beauty,
feeling and as yet unrealized affinity for the grandeurs of ancient Egyptian cul
ture. Belarusian philosopher Edward Soroko tried to determine what ideas were
used by Thutmose in the sculpture of Nefertiti [138]. He thinks that Thutmose’s
logic was very clear and simple. In ancient Egypt, harmony was the prerogative
of the Divine order that dominated the universe, and geometry was the main tool
of its expression. The Queen played the role of Goddess. Hence, her image, which
personified the wisdom of the world, must have been formed with geometrical
perfection and irreproachable harmony, beauty and clarity. As a matter of fact,
the main idea of ancient Egyptian aesthetic philosophy was to glorify the eternal,
the measured, and the perfect in a constantly changing universe.
Figure 1.38. Harmonic analysis of Nefertiti’s portrait
Soroko made a harmonious analysis of Nefertiti’s sculpture and came to the
conclusion that Thutmose had used the golden section principle as the basis of
its design. During his analysis, Soroko found a harmonious system of regular
geometric figures such as triangles, squares, and rhombi (see Fig. 1.38). Thus, he
established that the parts of these figures, if put in order according to their sizes,
are guided by one and the same ratio — the golden mean.
The metric structure of Nefertiti’s statuette shows that there were an
cient Egyptian sculptors consciously employing the principle of the golden
section in their creative work. Soroko’s analysis is one of the more compelling
confirmations of the role of the golden mean in ancient Egyptian art.
Chapter 1
The Golden Section
59
1.15. Conclusion
The geometrical problem of the “Division in Extreme and Mean Ratio”
(DEMR) came to us from Euclid’s Elements. It was named later the golden sec
tion, golden mean, golden number, golden proportion or divine proportion. DEMR
appears throughout Euclid’s Elements and was used by Euclid for the geometric
construction of the isosceles triangle with the angles 72°, 72° and 36° (the gold
en triangle), regular pentagon and dodecahedron. This fact may have given rise
to the widespread belief (see Proclus, commentator of Euclid’s Elements) that
the main goal of the Elements was to describe the geometric construction and
numerical interrelationship of the Platonic Solids. This means that the
Pythagorean doctrine about the numerical Harmony of the Spheres was real
ized in the greatest mathematical work of ancient science, Euclid’s Elements.
As the history of science shows, the golden section appears diffused with
in and throughout our cultural history. It became an aesthetic canon for the
Egyptian, Greek and Renaissance cultures. During many millennia the gold
en section was a subject of delight for the great scientists and thinkers includ
ing Pythagoras, Plato, Euclid, Leonardo da Vinci, Luca Pacioli, Johannes
Kepler, Allan Turing and countless others. A small list of some of the out
standing works of art and masterpieces which are based on the golden section
could contain Khufu’s Great Pyramid, the most famous of the Egyptian pyr
amids, Nefertiti’s sculptural portrait, the majority of Greek sculptural monu
ments, the magnificent Mona Lisa by Leonardo da Vinci, the works of Raphael,
Shishkin and Konstantin Vasiliev, Chopin’s etudes, the musical works of
Beethoven, Tchaikovsky and Bella Bartok, and the Моdulor of Corbusier.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
60
Chapter 2
Fibonacci and Lucas Numbers
2.1. Who was Fibonacci?
2.1.1. Leonardo of Pisa Fibonacci
Usually we associate the “Middle Ages” with the inquisition, forced conver
sions, burning of witches and heretics, and bloody Crusades in search of “Christ’s
Tomb.” Science was not in the spotlight. Under these circumstances, the appear
ance of the 1202 mathematical book “Liber abaci” by Italian mathematician Le
onardo of Pisa could not stay unnoticed by the scientific community. Who was
the author of this book, and why were his mathematical works so important for
Western European mathematics? To find the answers to these questions we have
to return to his time and reproduce the historical epoch, in which Leonardo of
Pisa, nicknamed Fibonacci, lived and worked.
It is worth noting that the period between the 11th and
12th centuries was an epoch of the brilliant flowering of
Arabian culture; however, this century was the beginning
of its downfall. At the end of the 11th century, before the
beginning of the Crusades, the Arabs were undoubtedly the
most educated people in the world and went far beyond
their Christian competitors. Arabian influence had pene
trated to the West long before the Crusades. After the Cru
Fibonacci
sades, Arabian culture continued to influence the West and (c. 1170after 1228)
this mingling of cultures began to erode the Arabian world. Western research
ers were dazzled with the Arabian world’s art and scientific achievements. Their
interest rapidly increased in Arabian geographical maps, algebra, astronomy tu
torials, and architecture. During this time the great European mathematician
of the Middle Ages, Leonardo of Pisa, nicknamed Fibonacci (son of Bonacci or
son of the bull), lived and worked in Pisa. He admired Arabic science and math.
Chapter 2
Fibonacci and Lucas Numbers
61
We know little about Fibonacci’s life. The exact date of his birth is un
known, though he is thought to have been born around 1170. His father was a
merchant and a government official who represented a new class of executives
generated by the “Commercial Revolution.” At that time the city of Pisa was
one of the largest commercial Italian centers that actively cooperated with
the Islamic East. Fibonacci’s father traded in one of the trading posts founded
by Italians on the northern coast of Africa. Thanks to that, he was able to give
his son Fibonacci, the future mathematician, a good mathematical education
in one of the Arabian educational institutions.
Moritz Cantor, wellknown historian of mathematics, called Fibonacci “the
brilliant meteor that flashed on the dark background of the Western Europe
an Middle Ages.”
Fibonacci wrote several mathematical works, including Liber Abaci, Liber
Quadratorum, and Practica Geometriae. The book Liber Abaci is the best known
of them. This book was printed twice during the life of Fibonacci, originally in
1202 and again in 1228. The book consisted of 15 sections covering the fol
lowing topics: the new Hindu numerals and representation of numbers with
their help (Section 1); multiplication, addition, subtraction and division of
numbers (Sections 25); multiplication, addition, subtraction and division of
fractions (Sections 67); finding the prices of the goods and their exchange,
the rules of mutual aid and the rule of the “double false situation” (Sections 8
13); finding the quadratic and cubic roots (Section 14); and the rules related
to geometry and algebra (Section 15).
The book was intended as a manual for traders, though its significance far
exceeded the bounds of trade practice. Fibonacci’s book is a sort of mathemati
cal encyclopedia of the Middle Ages epoch. Section 12 is of particular interest.
It makes up nearly one third of the book and, apparently, Fibonacci paid special
attention to it and demonstrated great innovation within it. The bestknown
Fibonacci problem is that of “rabbit reproduction.” Its solution purportedly re
sulted in the discovery of the numerical sequence 1, 1, 2, 3, 5, 8, 13 ..., later called
Fibonacci Numbers. This famous problem will be reviewed later.
2.1.2. Fibonacci and Abu Kamil
Roger HerzFischler [40] pointed out that Fibonacci borrowed many mathe
matical problems from the Arab mathematicians, in particular, Abu Kamil. How
ever, in the summary of this comparison HerzFischler did not accuse Fibonacci
of plagiarism. He wrote [40]: “Again when we turn to Fibonacci’s presentation of
the problems from Abu Kamil’s, On the Pentagon and Decagon, we find new meth
Alexey Stakhov
THE MATHEMATICS OF HARMONY
62
ods of solution that again display a deep understanding and ability. We must con
clude then that either there were several, now lost, works from which Fibonacci
obtained his material or he was responsible for significantly raising the level of the
applications of the properties of DEMR to various computational problems.”
2.1.3. Influence of Fibonacci’s Works on the Development of the
European Mathematics
Although Fibonacci was one of the great mathematical intellects in the histo
ry of Western European mathematics, his contribution in mathematics continues
to be understated. The Russian mathematician Professor Vasiliev, in his book In
teger Number (1919), noted the merits of Fibonacci’s creative mathematical work:
“The works of the educated merchant from Pisa were so far above and
beyond the level of mathematical knowledge of the scientists of those times,
that the influence of his work on mathematical literature only became recog
nized two centuries after his death, at the end of the 15th century, when many
of his theorems and problems were included by Luca Pacioli… and in the be
ginning of the 16th century, when the group of talented Italian mathemati
cians Ferro, Cardano, Tartalia, and Ferrari produced the beginnings of higher
algebra thanks to the solution of the cubic and biquadrate equations.”
It follows from this statement, that Fibonacci surpassed the Western Eu
ropean mathematicians of his time by almost two centuries. His historical role
for Western science is similar to the role of Pythagoras who acquired his sci
entific knowledge from Egyptian and Babylonian sciences and then transferred
them to Greek science. Fibonacci received his mathematical education in the
Arabian educational institutions and transferred the Arabic knowledge of math
to Western European science. Much of the knowledge acquired there, in par
ticular, the ArabicHindu decimal notation, was introduced by him to West
ern European mathematics. Thus, he provided the fundamentals for the fur
ther development of Western European mathematics.
2.2. Fibonacci’s Rabbits
2.2.1. The “Rabbit Reproduction Problem”
Fibonacci made a valuable contribution to science and the development of
mathematics, however, the irony of his fate is that in modern mathematics he is
Chapter 2
Fibonacci and Lucas Numbers
63
generally only known as the author of an unusual numerical sequence, the Fi
bonacci numbers. He deduced that sequence when he was looking for the solu
tion to the now famous rabbit reproduction problem. The formulation and solu
tion of this problem is considered to be Fibonacci’s main contribution to the
development of combinatorial analysis. In it Fibonacci anticipated the recur
sive method that was recognized as one of the most powerful methods of solving
combinatorial problems. The recursive relation achieved by Fibonacci is reput
ed to be the first recursive relation in mathematical history. Fibonacci formu
lated the core of the rabbit reproduction problem as follows:
“A pair of rabbits were placed within
an enclosure so as to determine how many
pairs of rabbits will be born there in one
year, it being assumed that every month a
pair of rabbits produces another pair, pro
vided that rabbits only begin to produce a
new pair at their maturity, which is two
Figure 2.1. Fibonacci’s rabbits
months after their own birth.”
For the solution to this problem (Fig. 2.1) we define a pair of the mature
rabbits by A, and a pair of the newborn rabbits by B. We display the process of
reproduction with two transitions that describe the rabbits’ monthly trans
formations:
A→AB
(2.1)
B→A
(2.2)
Note that the transition (2.1) simulates a monthly transformation of each
pair of mature rabbits into two pairs, namely, the same pair of mature rabbits
A and the newborn pair В. The transition (2.2) simulates the process of rabbit
maturation when a newborn pair B is transformed into a mature pair А. Fur
ther, by beginning with a mature pair A, the process of rabbit reproduction
can be represented by Table 2.1.
Table 2.1. Rabbit reproduction
Date
January, 1
February, 1
March, 1
April, 1
May, 1
June, 1
Pairs of rabbits
A
AB
ABA
ABAAB
ABAABABA
ABAABABAABAAB
A
1
1
2
3
5
8
B
0
1
1
2
3
5
A+B
1
2
3
5
8
13
Note that in the columns A, B and A+B of Table 2.1 we can see the num
bers, respectively, of mature (A), newborn (B) and total (A+B) rabbits.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
64
Studying the A, B and (A+B) sequences, we find the following regularity: each
number of the sequence is equal to the sum of the previous two numbers. If we now
designate the nth number of the sequence that satisfies the rule as Fn, then the
above general rule can be represented by the following mathematical formula:
Fn=Fn1+Fn2.
(2.3)
Such formula is called recurrence or recursive relation.
Note that the specific values of the numeric sequences generated by the
recursive relation (2.3) depend on the initial values (the seeds) of the sequence
F1 and F2. For example, we have F1 = F2 =1 for the A series and for this case the
recursive formula (2.3) generates the following numerical sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ....
(2.4)
For the B series we have: F1=0 and F2=1 then the corresponding numerical
sequence for this case is as follows:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ....
At last, for the (A+B) series we have: F1=1 and F2=2 then the correspond
ing numerical sequence for this case is the following:
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ....
The numerical sequence (2.4) is usually called Fibonacci numbers. Fi
bonacci numbers have a series of remarkable mathematical properties, which
we will describe further.
2.2.2. About the Rabbits
Why do rabbits occur in the history of mathematics? Rabbits are mammals of
the hare family. The countries of Spain, France, and Italy are considered to be the
native land of wild rabbits; from these countries the rabbits were introduced to
other countries. Wild rabbits now inhabit the southern and middle parts of West
ern Europe, and also Africa, Asia, Australia, New Zealand and America.
A special feature of rabbits is their surprising rate of reproduction. Female
rabbits become mature at the age of 34 months and are able to reproduce
throughout the year. A female rabbit’s pregnancy lasts for 2832 days (on the
average 30 days); this means, that, by considering process of rabbit reproduc
tion, Fibonacci employed biological facts. However, mature rabbits may pro
duce 8 to 10 newborn rabbits monthly. Thus, the rabbits reproduce more in
tensively than Fibonacci suggested in his famous problem.
This exclusive ability of rabbit reproduction can explain why many coun
tries consider the “rabbit invasion” to be a “national tragedy.” Australia is one of
Chapter 2
Fibonacci and Lucas Numbers
65
the examples. In 1837 one Australian farmer started a rabbit farm of 24 rabbits.
The rabbits that reproduced and escaped to freedom could have nearly destroyed
all the greenery of the continent. The Australian government succeeded in re
ducing the number of rabbits by its resolute measures in the struggle with the
socalled “longeared locust.” In Australia, war was declared against the rabbits
and has lasted more than 150 years. Throughout the continent, Australians had
constructed a kind of “Great Wall of China” of many hundreds of kilometers, an
insuperable barrier for the rabbits. The war has progressed with variable suc
cess. Wild Australian rabbits have learned to climb trees, becoming terrifyingly
aggressive, capable of attacking the fields and kitchen gardens of the farmers.
The “prolific tribe of rabbits,” which had influenced the famous Italian math
ematician, now took the Italian island Ustina (to the North of Sicily) under
siege. 100,000 rabbits overran the 1000 inhabitants of this small island. In con
trast to the Australian inhabitants the native population of Ustina yielded to
the rabbits without a fight; already one fifth of the inhabitants have emigrated
from the island.
It is necessary to remember the flip side of the “rabbit problem”; rabbit
meat is considered useful and tasty. Italy, Fibonacci’s native land, is one of the
largest producers of rabbit meat. Remembering this, the Italian (and nonItal
ian) historians of mathematics should work to derive the answer to the fol
lowing question: what was the main reason for Fibonacci introducing rabbit
reproduction into mathematics: a love for mathematics or rabbit meat?
2.2.3. Dudeney’s Cows
The English puzzlist Henry E. Dudeney (18571930) wrote several excel
lent books on puzzles. In one of them he adapted Fibonacci’s rabbits to cows
by making Fibonacci’s problem more realistic. He replaced months by years
and rabbits with bulls (males) and cows (females). Dudeney states his cow
problem as follows:
If a cow produces its first shecalf at age of two and after that produces
another single shecalf every year, how many shecalves are there after 12 years,
assuming none die?
In Dudeney’s opinion, this simplifies the problem and makes it quite realistic.
2.2.4. Honeybees and Family Trees
As known, Fibonacci took a physical problem and simplified it in the way
many mathematicians often do at first, namely, he simplified the problem and
then observed to see what would happen. This series has many interesting
Alexey Stakhov
THE MATHEMATICS OF HARMONY
66
and useful applications, which we will see later. Now let us consider another
reallife situation, honeybee reproduction, which can also be precisely mod
eled on the Fibonacci series
There are over 30,000 species of bees and most of them live solitary lives.
The honeybees are the best known among them. They usually live in a colony
called a hive and have an unusual “Family Tree.” In fact, there are many surpris
ing features of the honeybees and, in this section, we will show how the Fi
bonacci numbers can be used to construct the genealogical trees of honeybees.
Before we discuss the Fibonacci series and honeybees, let us look at an unusu
al fact about them. Not all honeybees have two parents! In a colony of honeybees
there is one special bee called the queen bee. There are many workerbees, who
are female too, but unlike the queen bee, they do not produce eggs. There are
some male bees called dronebees. Dronebees are produced by the unfertilized
eggs of the queen bee, so the male bees have only a mother, and no father. All
female bees are produced when the queen
bee has mated with a male bee and so fe
male bees have two parents (Fig. 2.2). The
females usually become workerbees.
However, when some of them are fed with
a special substance called royal jelly, each
grows into a queen bee ready to go off to
start a new colony. When this occurs, the
Figure 2.2. Family tree of honeybees
bees form a swarm and leave their hive in
search of a place to build a new nest.
Here we follow the convention of “Family Trees” where parents appear above
their children, so the latest generations are at the bottom, and the higher up we
go the older the generations. Such trees place all ancestors of the descendent at
the top of the diagram. We would get quite a different tree if we listed all the
descendants (offspring) of an ancestor in a way similar to the rabbit problem,
where we placed all descendants of the initial pair at the bottom.
Now, let us look at the family tree of the malebee.
1. The male bee has 1 parent, the queen bee.
2. The male bee has 2 grandparents because his mother (the queen bee)
had two parents, a male and a female.
3. The male bee has 3 greatgrandparents: i.e. his grandmother had two
parents; however, his grandfather had only one parent.
4. How many greatgreatgrand parents did the malebee have?
It is possible to make the following table, which sets forth a family tree of
each male bee and each female bee.
Chapter 2
Fibonacci and Lucas Numbers
67
It follows from Table 2.2 that the reproduction of honeybees is carried out
according to “Fibonacci’s principle!”
Table 2.2. Fibonacci numbers in a family tree of honeybees
Number of
malebees
Number of
female
bees
Number
of
parents
Number of
grand
parents
Number of
greatgrand
parents
Number of
greatgreat
grand parents
Number of
greatgreat
greatgrand
parents
1
2
3
5
8
2
3
5
8
13
2.3. Numerology and Fibonacci Numbers
2.3.1. Some Mathematical Properties of Numerological Values
Increasingly popular today, numerology is an ancient tradition that attract
ed attention of civilization’s greatest minds. There is evidence that numerology
was in use in China, Greece, Rome, and Egypt for a long time before Pythago
ras, who is generally considered to be the “father” of numerology.
Numerology, not unlike astrology and other esoteric subjects, has experi
enced a revival in recent years. Many modern numerologists have adopted
the original Pythagorean system. It is a simple system that assigns a number
value (from one to nine) to every letter of the alphabet: A is 1, B is 2, and so
on. According to traditional summation, we add for example, 7+8=15. In nu
merology, this is not enough: we use a further method of summation known as
the “Fadic” system, or “natural summation.” This simply means we sum two
or more digits together until we arrive at a single digit. For our example, 7 and
8 add up to 15, however, in numerology we sum 1+5, for a final answer 6.
Despite its esoteric aspects, numerology involves a deep mathematical
conception. In fact, a calculation of the numerological values of one or anoth
er number from the mathematical point of view is very close to obtaining the
remainder of some number through division by 9. In mathematics such oper
ation is called a reduction by modulo 9.
Let us denote a numerological value of the integer a by k(a). We say that
two integers a and b are comparable by modulo m (m being a natural number)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
68
if at the division by m they result in the same remainder. In other words, a and
b are comparable by modulo m if their remainder a and b are identical.
Example: 32 and 39 are comparable by modulo 7 because 32=7×4+4,
39=7×5+4, both having the same remainder: 4.
The statement “a and b are comparable by modulo m” is written in the form:
a≡b (mod m).
Note that there is only one distinction between the numerological value
of the number a and its remainder by modulo m=9. Consider, for example,
the Fibonacci number of 144. As this number is perfectly divisible by 9, 144≡0
(mod 9) and the numerological value of the number 144 is equal to 9, i.e.
k(144)=1+4+4=9.
A relation of comparison by the modulo m has many properties similar to
the properties of traditional integers. For example, if
a1≡b1 (mod m) and a2≡b2 (mod m),
then we have
a1a2≡b1b2 (mod m)
and
(2.5)
a1+a2≡(b1+b2)(mod m).
As 10≡1 (mod 9) then by using (2.5) it is easy to prove that
(2.6)
(10i)≡1 (mod 9).
Now, let us represent a natural number N in the decimal system:
(2.7)
(2.8)
N = bn 10n −1 + bn −110n −2 + ... + bi 10i −1 + ... + b2 101 + b1100 ,
where bn,bn1,…,bi,…,b2,b1 are decimal numerals of the number N that take their
values from the set of the decimal numerals {0,1,2,3,4,5,6,7,8,9}.
The abridged notation of the sum (2.8) N= bn,bn1,…,bi,…,b2,b1 is called a
decimal notation of the number N.
By using (2.5) through (2.8), we can write:
k ( N ) = k ( bn + bn −1 + ... + bi + ... + b2 + b1 ) .
If we represent the sum
(2.9)
S = bn + bn −1 + ... + bi + ... + b2 + b1
in the decimal system (2.8)
S = d m 10m −1 + d m −110m −2 + ... + d j 10 j −1 + ... + d2 101 + d1100 ,
(2.10)
then we can obtain a new expression for the numerological value of the
number N:
(
)
k ( N ) = k dm + dm −1 + ... + d j + ... + d2 + d1 .
(2.11)
Chapter 2
Fibonacci and Lucas Numbers
69
This procedure continues until the last sum of the kind (2.11) is less than
the number 10. It is the numerological value of the number N.
From the above examination, it is easy to deduce a number of mathemat
ical properties of the “numerological values.” First of all, we can find a numer
ological value of the sum of two numbers, for example, N1+N2. For this pur
pose, we represent the numbers N1 and N2 in the decimal notation (2.8):
(2.12)
N 2 = d n 10n −1 + d n −110n − 2 + ... + d i 10i −1 + ... + d2 101 + d1100
Then, we can represent the sum N1+N2 as follows:
N1 + N 2 = cn 10n −1 + cn −110n −2 + ... + ci 10i −1 + ... + c2 101 + c1100 
(2.13)
(2.14)
+  dn 10n −1 + d n −110n −2 + ... + d i 10i + ... + d2 10i −1 + d1100  .
It follows from (2.14) that the numerological value of the sum N1+N2 is
calculated according to the expression:
k ( N 1 + N 2 ) = k ( cn + cn −1 + ... + ci + ... + c2 + c1 ) + dm + dm −1 + ... + d j + ... + d2 + d1 
(2.15)
= k ( cn + cn −1 + ... + ci + ... + c2 + c1 ) + k d m + d m −1 + ... + d j + ... + d2 + d1 .
(
(
)
)
We can write the expression (2.15) as follows:
k ( N1 + N 2 ) = k ( N1 ) + k ( N 2 ) .
(2.16)
Thus, the numerological value of the sum N1+N2 is equal to the sum of the
numerological values of the initial numbers N1 and N2 . As examples, we calcu
late the numerological value of the sum:
17711+5702887=5720598.
It is clear that the numerological value of the numbers 17711, 5702887
and 5720598 are equal, respectively,
k(17711)=1+7+7+1+1=17, 1+7=8;
k(5702887)=5+7+0+2+8+8+7=37, 3+7=10, 1+0=1;
k(5720598)= 5+7+2+0+5+9+8=36, 3+6=9.
On the other hand, we have
k(5720598)=k(17711)+k(5702887)=8+1=9.
2.3.2. Fibonacci Numerological Series
Let us now examine the Fibonacci numbers from numerological points of
view. For this purpose we write the first 48 Fibonacci numbers together with
their numerological values k(F1) through k(F48) (Table 2.3).
THE MATHEMATICS OF HARMONY
Alexey Stakhov
70
Table 2.3. Numerological values of Fibonacci numbers
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Fn
1
1
2
3
5
8
13
21
34
55
89
144
233
377
610
987
1597
2584
4181
6765
10946
17711
28657
46368
k(Fn)
n
Fn
k(Fn)
1
1
2
3
5
8
4
3
7
1
8
9
8
8
7
6
4
1
5
6
2
8
1
9
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
75025
121393
196418
317811
514229
832040
1346269
2178309
3524578
5702887
9227465
14930352
24157817
39088169
63245986
102334155
165580141
267914296
433494437
701408733
1134903170
1836311903
2971215073
4807526976
1
1
2
3
5
8
4
3
7
1
8
9
8
8
7
6
4
1
5
6
2
8
1
9
Let us consider the recursive relation (2.3) for Fibonacci numbers. If we
use a general identity (2.16), we can write the following recursive relation for
the numerological values of the adjacent Fibonacci numbers:
k ( Fn ) = k ( Fn −1 ) + k ( Fn −2 ) .
(2.17)
We can see from Table 2.3 that this regularity is valid for all Fibonacci num
bers in that table. For example, by using Table 2.3, we can write: k(F41) = 4. We
can also calculate the numerological value of the Fibonacci number F41 by using
the recursive relation (2.17). In fact,
k ( F41 ) = k ( F40 ) + k ( F39 ) = 6 + 7, 1 + 3 = 4.
Analysis of the numerological values k(F1) to k(F48) lead us to an unexpect
ed result. Beginning with the Fibonacci number F25, the numerological values
start to recur, that is, the numerological values k(F25) to k(F48) are a repetition of
the numerological values k(F1) to k(F48). We can continue Table 2.3 in order to
be convinced that the numerological values k(F49) to k(F72) are a repetition of
k(F1) to k(F24) and k(F25) to k(F48). Thus, the Fibonacci Numerological Series k(Fi)
(i=1, 2, 3, ...) is a periodic sequence with the period of length 24:
Chapter 2
Fibonacci and Lucas Numbers
71
1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9
(2.18)
Let us divide the period (2.18) into two parts (with 12 numbers in each part):
1
1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9
2
8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9
and then sum them in pairs. As a result, we obtain the regularity presented in
Table 2.4.
It follows from Table 2.4 that the sum of the first 11 pairs of the Fibonacci
numerological series is equal to the number 9. However, the sum of the last,
that is, the 12th pair is equal to 9+9 = 18; note that the numerological value of
this last sum is also equal to 9.
If we sum all numbers of the first period, then this sum is equal to
1+1+2+3+5+8+4+3+7+1+8+9+8+8+7+6+4+1+5+6+2+8+1+9 = 117.
Table 2.4. Regularity
The numerological value of the sum 117 is again
of the Fibonacci
equal to 9. This means that the numerological value of
numerological series
the sum of the first 24 Fibonacci numbers is equal to 9.
By analogy, we can assert that the numerological sum
1 + 8 = 9
1 + 8 = 9
of the next 24 Fibonacci numbers will also be equal to
2 + 7 = 9
9. Thus, if we begin from the first Fibonacci numbers F1
3 + 6 = 9
to F24 and examine the subsequent periods of 24 Fi
5 + 4 = 9
bonacci Numbers, for example, F25 to F48, F49 to F72, and
8 + 1 = 9
4 + 5 = 9
so on, then we will find that the numerological values
3 + 6 = 9
of these sums are each equal to 9. In order to calculate
7 + 2 = 9
the numerological value of the Fibonacci series, we
1 + 8 = 9
should calculate the numerological value of the follow
8 + 1 = 9
9 + 9 = 18
ing infinite sum 9+9+…+9+…. It is easy to prove that
the numerological value of this sum is equal to 9. This
reasoning results in the following theorem.
Theorem 2.1. The Fibonacci numerological series has the period (2.18) of
length 24; here the numerological value of the sum of the first 24 Fibonacci num
bers and all subsequent periods of 24 Fibonacci numbers are each equal to 9.
This means that the number 9 is a “numerological essence” of the Fi
bonacci series, that is, this number expresses a “sacred” property of the
Fibonacci series.
2.3.3. Another Periodic Properties of Fibonacci Numbers
It is necessary to point out that the period (2.18) that appears in the nu
merological study of Fibonacci numbers is not the only example of a periodic
THE MATHEMATICS OF HARMONY
Alexey Stakhov
72
property of Fibonacci numbers. For example, we can calculate the values of
the Fibonacci series by Mod 2 and Mod 3 (see Table 2.5).
Table 2.5. Periodicities of Fibonacci numbers taken by Mod 2 and Mod 3
Fn
1
1
2
3
5
8
13
21
34
55
89 144 233 377 610 987
Mod 2 1
1
0
1
1
0
1
1
0
1
1
0
1
1
0
1
Mod 3 1
1
2
0
2
2
1
0
1
1
2
0
2
2
1
0
We can see that the Fibonacci numbers by Mod 2 have a period 1, 1, 0 of
length 3 and by Mod 3 a period 1, 1, 2, 0, 2, 2, 1, 0 of length 8. By studying the
numerical sequences obtained by taking Fibonacci series by Mod k, the Amer
Table 2.6. Periodicities of Fibonacci numbers by different k module
Modulo k
Length of
the period
2
3
4
5
6
7
8
9
12
16
3
8
6
20
24
16
12
24
24
24
ican scientist Jay Kappraff found the following periodicities of such sequenc
es as given in Table 2.6.
Note that mathematicians discovered many interesting numbertheoretic
properties of Fibonacci numbers concerning their divisibility. Consider the
following remarkable properties of Fibonacci numbers proved in [13]:
1. If n is divisible by m, then Fn is divisible by Fm. For example, consider
two Fibonacci numbers, F26=121393 and F13=233. We can see that F26 is divis
ible by F13 (121393:233=521) because 26 is divisible by 13 (26:13=2).
2. If Fn≡0 (mod k) and Fm≡0 (modk) then also Fn=m≡0 (mod k) and
Fnm≡0 (mod k) (for n>m). For example, consider two Fibonacci numbers,
F24=46368 and F12=144. We can calculate that F24≡0 (mod 9) and F12≡0 (mod 9);
then for the Fibonacci numbers F24+12=F36=14930352 and F2412=F12=144 we
have: F36≡0 (mod 9) and F12≡0 (mod 9).
A surprising periodicity of the Fibonacci series taken by Mod k (see Ta
bles 2.4 2.6) produces a feeling of rhythm and harmony hidden in this re
markable numerical sequence!
2.4. Variations on Fibonacci Theme
The variations on a given theme in music are known as genre. A distinctive
feature of the musical works of various genres consists of the fact that they be
Chapter 2
Fibonacci and Lucas Numbers
73
gin, in most cases, with one simple essential musical theme, which thereinafter
undergoes considerable changes of tempo, mood and nature. However, no mat
ter how extreme the variations are, the listeners are absolutely impressed that
each variation is a natural development of the main theme.
If we follow the example of musical genre and select a simple mathemati
cal subject, such as the Fibonacci series, we can consider this series together
with its numerous variations.
2.4.1. Formulas for the Sums of Fibonacci Numbers
Fibonacci numbers have a number of delightful mathematical properties
that have stimulated the imaginations of mathematicians over the centuries.
Let us calculate, for example, the sum of the first n Fibonacci numbers. Begin
from the simplest sums:
1+1 = 2 = 3 −1
1+1+ 2 = 4 = 5 −1
1+1+ 2 + 3 = 7 = 8 −1
1 + 1 + 2 + 3 + 5 = 12 = 13 − 1
(2.19)
If we consider in the sums (2.19) the numbers marked by bold type: 3, 5, 8,
13, …, then it is easy to see that they are Fibonacci numbers! Then, we can
write the sums (2.19) as follows:
F1 + F2 = F4 − 1; F1 + F2 + F3 = F5 − 1; F1 + F2 + F3 + F4 = F6 − 1.
It is clear that the general formula has the following form:
(2.20)
F1 + F2 + ... + Fn = Fn + 2 − 1.
Now, let us consider the sum of the n sequential Fibonacci numbers with
the odd indexes 1,3,5,…,2n1,…. To this end we start from the simplest sums:
1+ 2 = 3
1+ 2 + 5 = 8
1 + 2 + 5 + 13 = 21
(2.21)
1 + 2 + 5 + 13 + 34 = 55
By analyzing (2.21), we find the following regularity for Fibonacci num
bers: the sum of the n sequential Fibonacci numbers with the odd indexes
is equal to Fibonacci numbers! In general, the partial sums (2.21) can be
written in the form of the following general identity:
F1 + F3 + F5 + ... + F2n −1 = F2n .
(2.22)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
74
It is easy to prove the similar formulas for the sums of the n sequential
Fibonacci numbers with even indexes:
(2.23)
F2 + F4 + F6 + ... + F2n = F2n +1 − 1.
Now, let us find the sum of the squares of the n sequential Fibonacci
numbers:
F12 + F22 + ... + Fn2 .
Start from the analysis of the simplest sums of the kind (2.24):
(2.24)
12 + 12 = 2 = 1 × 2
12 + 12 + 22 = 6 = 2 × 3
12 + 12 + 22 + 32 = 15 = 3 × 5
(2.25)
12 + 12 + 22 + 32 + 52 = 40 = 5 × 8
The analysis of (2.25) results in the following general formula:
(2.26)
F12 + F22 + ... + Fn2 = Fn Fn +1 ,
that is, the sum of squares of n sequential Fibonacci numbers is equal to
the product of the greatest Fibonacci number used in this sum multiplied
by the next Fibonacci number!
Also we can find the sum of the squares of two adjacent Fibonacci
numbers:
Fn2 + Fn2+1 .
(2.27)
Start from the analysis of the simplest sums of the kind (2.27):
1 + 12 = 1 + 1
2
= 2
12 + 22 = 1 + 4 = 5
22 + 32 = 4 + 9 = 13
(2.28)
32 + 52 = 9 + 25 = 34
By analyzing (2.28), we can find another remarkable regularity: the sum
of squares of two adjacent Fibonacci numbers is always equal to a Fibonacci
number! The general form of this regularity can be expressed as follows:
Fn2 + Fn2+1 = F2n +1 .
(2.29)
2.4.2. Connection of Fibonacci Numbers to the Golden Mean
Now, let us show a connection of Fibonacci numbers to the golden mean.
With this purpose in mind, we can examine a sequence of fractions that are
built up by the ratios of adjacent Fibonacci numbers:
Chapter 2
Fibonacci and Lucas Numbers
75
1 2 3 5 8 13 21 34
,
,
,
,
,
,
,
, ...
(2.30)
1 1 2 3 5
8 13 21
The first terms of the sequence (2.30) have the following values:
1
2
3
5
8
13
21
= 1; = 2; = 1.5; = 1.666; = 1.6;
= 1.625;
= 1.615; ...
1
1
2
3
5
8
13
The question is: what is the limit of the sequence (2.30) if we direct the index
n to infinity? To answer this question, let us consider the representation of the
golden mean in the form (1.15). It is easy to prove that the sequence (2.30) is
connected directly with the representation (1.15). In fact, the fractions (2.30) are
sequential approximations of the continuous fraction (1.15), namely:
1
=1
1
(the first approximation);
2
1
=1+
1
1
(the second approximation);
3
1
=1+
1
2
1+
1
(the third approximation);
5
=1+
3
1
1+
(the fourth approximation).
1
1+
1
1
If we continue this process to infinity, we obtain:
lim
Fn
n →∞ F
n −1
=τ=
1+ 5
.
2
(2.31)
The result given by the expression (2.31) is the “key” result for our re
search because it shows a deep connection between the Fibonacci numbers
and the golden mean. This means, that like the golden mean itself, Fibonacci
numbers express harmony in the world around us!
2.4.3. “Iron Table” by Steinhaus
Renowned Polish mathematician Steinhaus constructed a table of ran
dom numbers by using the golden mean. For this purpose, he multiplied
10 000 integers from 1 to 10 000 by the number ϕ=τ1=0.61803398, where τ is
Alexey Stakhov
THE MATHEMATICS OF HARMONY
76
the golden mean. As a result, he obtained the sequence of numbers multiplied
by ϕ, that is:
1ϕ, 2ϕ 3ϕ, ... , 4181ϕ, ... , 6765ϕ, ... , 10000.
Steinhaus called this numerical sequence the Golden Numbers. Each
Golden Number consists of integer and fractional parts. For example, the
number 1000ϕ = 618.03398 has the integer part 618 and the fractional part
0.03398. The number 4181ϕ=2584.00007 has the integer part 2584 and
the fractional part 0.0007, and so on. Moreover, neither a Golden Number
with fractional part equal to 0 exists nor do two Golden Numbers with the
equal fractional parts. This means that every Golden Number has a unique
fractional part.
If we put the Golden Numbers in order in a special table according to their
increasing fractional parts, we find that the number 4181ϕ has the least frac
tional part and, therefore, this Golden Number should start this table; also we
find that the number 6765ϕ has the largest fractional part and, therefore, this
Golden Number has to end the table:
4181
3194
……
8739
0987
8362
7365
……
1974
5168
1597
0610
……
6155
9349
5778
4791
……
3571
2584
9959
8972
……
7752
6765
Steinhaus named this the Iron Table taking into consideration some of its
unique properties. The Iron Table demonstrates deep connections with Fibonac
ci numbers. The first property is that a difference between the adjacent Gold
en Numbers of the Iron Table by absolute value is always equal to one of three
numbers: 4181, 6765 and 2584. In fact, we have:
8362 − 4181 = 4181, 8362 − 1597 = 6765, 5778 − 1597 = 4181, ...
9349 − 5168 = 4181, 9349 − 2584 = 6765, 6765 − 2584 = 4181.
It is easy to find the numbers of 2584, 4181 and 6765 if we continue the
Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 897, 1597, 2584, 4181, 6765,...
Hence, the numbers 2584, 4181, 6765 are three adjacent Fibonacci
numbers:
F18 = 2584, F19 = 4181, F20 = 6765.
We can see that the Iron Table starts with the Fibonacci number F18=2584
and ends with the Fibonacci numbers F19=4181 and F20=6765.
Chapter 2
Fibonacci and Lucas Numbers
77
It is clear that the Iron Table may be constructed for any arbitrary natural
number N. Polish scientist Jan Grzedzielski in his book EnergyGeometric Code
of Nature [26] analyzed the Iron Tables for the cases N=Fn where Fn is Fibonac
ci number. He obtained an interesting regularity that appears at the conver
sion of the Iron Table with N=Fn1 into the next Iron Table with N=Fn. This
latter table is constructed from the preceding table (with N=Fn1) by means of
the disposition of new numbers Fn1+1, Fn2+2,…,Fn1,Fn on particular positions
in the new Iron Table. In Grzedzielski’s opinion, this method of Iron Table con
struction “resembles the functioning of all radiation spectra in Nature.”
2.5. Lucas Numbers
2.5.1. FrancoisEdouardAnatole Lucas
Fibonacci did not continue to study the mathematical properties of the nu
merical series (2.4). However, the study of Fibonacci numbers was continued
by other mathematicians. Since the 19th century, the mathematical works de
voted to Fibonacci numbers, according to the witty expression of one mathe
matician, “began to be reproduced like Fibonacci’s rabbits.” The French math
ematician Lucas became one of the leaders of this research in the 19th century.
What do we know about Lucas? The French math
ematician FrancoisEdouardAnatole Lucas was born
in 1842. He died in 1891 as a result of an accident that
occurred at a banquet, when a dish was smashed and a
splinter wounded his cheek. Lucas died from an infec
tion some days later.
Lucas’ major works fall in the area of number
theory and indeterminate analysis. In 1878 Lucas
gave the criterion for the determination of the “pri
mality” of Mersenne numbers of the kind Mp=2p1.
FrancoisEdouard
Anatole Lucas
Applying his own method, Lucas proved that the
(18421891)
Mersenne number
M127 = 2127 − 1 = 170141183460469231731687303715884105727
is prime. For 75 years, this number was the greatest prime number known in
mathematics. He also found the 12th “perfect number” and formulated a num
ber of interesting mathematical problems.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
78
We can give some explanation for Lucas’ scientific achievements. It is well
known that the prime numbers are divisible only by 1 and itself. The first few
prime numbers are: 2, 3, 5, 7, 11, 13, … . Already the Pythagoreans knew that
the number of prime numbers is infinite (the proof of this fact is in The Ele
ments of Euclid). A study of the prime numbers and determination of their
distribution in the natural series is a rather difficult problem in number theo
ry. Therefore, the scientific result that was obtained by Lucas in the field of
prime numbers, undoubtedly, belonged to the category of outstanding mathe
matical achievements.
It is curious, that Lucas took a great interest in the socalled Perfect Num
bers. What are Perfect Numbers? As is well known, the Pythagorean theory of
numbers had a qualitative character, that is, the Pythagoreans were interest
ed in the qualitative aspects of numbers. They attributed to numbers some
unusual properties. In this connection, the socalled Perfect Numbers are of
special interest. For example, consider the number 6. Its feature is that this
number is equal to the sum of its divisors, that is, 6=1+2+3. Besides the num
ber 6, the Pythagoreans knew another two Perfect Numbers, 28 and 496:
28=1+2+4+7+14; 496=1+2+4+8+16+31+62+124+248.
The fourth perfect number is 8128. It was also known by ancient math
ematicians.
It is proven, that in the process of movement along natural series the
Perfect Numbers are found less frequently. Only four Perfect Numbers (4, 28,
496 and 8128) are found in the first 10,000 numbers of the natural series.
The search for Perfect Numbers became a fascinating passion for many math
ematicians. The fifth Perfect Number 212(2131) was found in the 15th centu
ry by the German mathematician Regiomontan. In the 16th century, the
German scientist Sheybel found two new Perfect Numbers: 8589869056 and
137438691328. Lucas in the 19th century found the twelfth Perfect Number.
Research in this area continues today where all the power of modern com
puters is being used. For example, the 18th Perfect Number that was found
by means of computer modeling has 2000 decimal digits.
In honor of Lucas, it is necessary to note one more of his scientific predic
tions. Already in the 19th century, long before the occurrence of modern com
puters, Lucas paid attention to technical advantages of the binary notation
for technical realization of computers and machines. This means that he an
ticipated by a century the outstanding American physicist and mathemati
cian John von Neumann’s preference for the binary system in the technical
realization of electronic computers (John von Neumann’s principles).
Chapter 2
Fibonacci and Lucas Numbers
79
2.5.2. Some Properties of Lucas Numbers
However, it is most relevant for our book that in the 19th century Lucas
attracted the attention of mathematicians to the remarkable numeric sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, …, which he named Fibonacci Numbers. In addition,
Lucas introduced the concept of generalized Fibonacci numbers that are cal
culated according to the following general recursive relation:
Gn = Gn −1 + Gn −2
(2.32)
at the initial terms (seeds) G1 and G2.
For example, the sequence of numbers 2, 8, 10, 18, 28, 46, … falls into the
generalized class of Fibonacci numbers satisfying the recursive relation (2.5)
at the seeds G1=2 and G2=8.
However, the main numerical sequence of the type (2.32), introduced by
Lucas in the 19th century, is a numerical sequence given by the following re
cursive relation:
Ln = Ln −1 + Ln −2
(2.33)
at the seeds
L1 = 1, L2 = 3.
(2.34)
Then, by using the recursive relation (2.33) at the seeds (2.34) we can
calculate the following numerical sequence called Lucas Numbers:
1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, ….
(2.35)
If we make the reasonings for Lucas numbers similar to Fibonacci num
bers, we can prove the following identities:
L1 + L2 + ... + Ln = Ln + 2 − 3
L1 + L3 + L5 + ... + L2n −1 = L2n − 2
L1 + L3 + L5 + ... + L2n −1 = L2n − 2
L21 + L22 + L23 + ... + L2n = Ln Ln +1 − 2
L2n + L2n +1 = 5F2n +1
lim
Ln
n →∞ L
n −1
=τ=
1+ 5
2
(2.36)
(2.37)
2.5.3. The “Extended” Fibonacci and Lucas Numbers
Up until now, we have studied Fibonacci and Lucas series Fn and Ln with
the positive indices n, that is, n=1,2,3,…. However, they can be extended with
Alexey Stakhov
THE MATHEMATICS OF HARMONY
80
the negative values of the indices n, that is, when the indices n take their val
ues from the set: n=0,1,2,3,….
The “extended” Fibonacci and Lucas numbers are represented in Table 2.7.
It follows from
Table 2.7. “Extended” Fibonacci and Lucas numbers
Table 2.7 that the
0
1
2
3
4
5
6
7
8
9
10
n
elements of the “ex
0
1
1
2
3
5
8
13 21 34 55
Fn
tended” numerical
0
1
1
2
3
5
8
13
21
34
55
Fn
sequences F n and
1
3
4
7
11 18 29 47 76 123
Ln have a number of
Ln 2
remarkable mathe
1
3
4
7 11 18 29 47 76 123
Ln 2
matical properties.
For example, for the odd indices n=2k+1 the elements of the sequences Fn and
Fn coincide, that is, F2k+1= F2k1, and for the even indices n=2k they are oppo
site by a sign, that is, F2k=F2k. For the Lucas numbers Ln all is vice versa, that
is, L2k=L2k; L2k+1= L2k1.
Now, let us consider the numerical sequences given in Table 2.7. Take, for
example, the Lucas number L4=7 and compare it with Fibonacci numbers. It
is easy to find that L4=7=5+2. However, the numbers 2 and 5 are the Fibonac
ci numbers F3=2 and F5=5.
Is this a mere coincidence? By continuing examination of Table 2.7, we
can find the following correlations that connect Fibonacci and Lucas num
bers: 1=0+1, 3=1+2, 4=1+3, 7=2+5, 11=3+8, 18=5+13, 29=8+21, and so on.
Now, let us compare the numerical sequences Ln and Fn. Here we find the
same regularity, that is, 1=0+(1), 3=1+2, 4=(1)+(3), and so on. Thus, we
found the following surprising mathematical identity that connects Lucas and
Fibonacci numbers:
(2.38)
Ln = Fn + Fn +1 ,
where the index n takes the following values: n= 0, ±1, ±2, ±3, ….
If we continue examination of Table 2.7, we can find other interesting iden
tities for Fibonacci and Lucas numbers:
Ln = Fn + 2Fn −1
(2.39)
(2.40)
Ln + Fn = 2Fn −1 .
Note that the above formulas for Fibonacci and Lucas numbers can be
considered to be the “golden treasure” of mathematics! And, by comprehend
ing these formulas, we can understand the delight of many outstanding math
ematicians of the 20th century, in particular, the Russian mathematician Ni
kolay Vorobyov and the American mathematician Verner Hoggatt. It was Pro
fessor Nikolay Vorobyov who was the author of the remarkable brochure
Chapter 2
Fibonacci and Lucas Numbers
81
Fibonacci Numbers [13] that became a mathematical bestseller of the 20th cen
tury. Professor Verner Hoggat was the founder of the Fibonacci Association,
the mathematical journal The Fibonacci Quarterly and the author of the book
Fibonacci and Lucas Numbers [16]. They saw in Fibonacci and Lucas numbers
a “mathematical secret of nature.” The desire to uncover this “secret” inspired
them to study these unique mathematical sequences!
2.5.4. Other Remarkable Identities for Fibonacci and Lucas Numbers
The next group of formulas is based on the following identity that connects
the generalized Fibonacci numbers Gn with the classical Fibonacci numbers Fn:
Gn + m = Fm −1Gn + FmGn +1 .
(2.41)
We can prove this formula by induction on m. For the cases m=1 and m=2
this formula is valid because
Gn +1 = F0Gn + F1Gn +1 = Gn +1 and Gn + 2 = F1Gn + F2Gn +1 = Gn + Gn +1 .
The basis of the induction is proved.
Let us suppose that the formula (2.41) is valid for the cases m=k and m=k+1,
that is,
Gn + k = Fk −1Gn + FkGn +1 , Gn + k +1 = FkGn + Fk +1Gn +1 .
By summarizing these formulas termwise, we obtain the following identity:
Gn + k + 2 = Fk +1Gn + Fk + 2Gn +1 .
The identity (2.41) is proved.
A number of interesting identities for Fibonacci and Lucas numbers fol
low from the identity (2.41). Suppose that Gi=Fi and m=n+1. Then, the iden
tity (2.41) is reduced to the following:
F2n +1 = Fn2+1 + Fn2 .
(2.42)
For the case Gi=Fi and m=n, the formula (2.41) is reduced to the following:
F2n = Fn −1 Fn + Fn Fn +1 = Fn ( Fn −1 + Fn +1 ) = Fn Ln .
(2.43)
By using the formula (2.43), we can write:
Fn +1 Ln +1 − Fn Ln = F2n + 2 − F2n = Fn Ln .
(2.44)
2.5.5. Lucas Numbers and Numerology
As was mentioned above, the Fibonacci series generates a periodic Fibonac
ci numerological series with the period (2.18). There is a question: whether
there is a similar periodicity for the Lucas series? Table 2.8 presents 48 Lucas
numbers and their numerological values.
THE MATHEMATICS OF HARMONY
Alexey Stakhov
82
Table 2.8. Numerological values of Lucas numbers
n
Ln
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
1
3
4
7
11
18
29
47
76
123
199
322
521
843
1364
2207
3571
5778
9349
15127
24476
39603
64079
103682
k(Ln)
n
Ln
k(Ln)
1
3
4
7
2
9
2
2
4
6
1
7
8
6
5
2
7
9
7
7
5
3
8
2
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
167761
271443
439204
710647
1149851
1860498
3010349
4870847
7881196
12752043
20633239
33385282
54018521
87403803
141422324
228826127
370248451
599074578
969323029
1568397607
2537720636
4106118243
6643838879
10749957122
1
3
4
7
2
9
2
2
4
6
1
7
8
6
5
2
7
9
7
7
5
3
8
2
The analysis of Table 2.8 shows that the numerological Lucas series
{k(Ln, n=1, 2, 3,…)} possess mathematical properties similar to the numerolog
ical Fibonacci series, that is, the numerological Lucas series is periodic with
the period of length 24 of the following kind:
1, 3, 4, 7, 2, 9, 2, 2, 4, 6, 1, 7, 8, 6, 5, 2, 7, 9, 7, 7, 5, 3, 8, 2
(2.45)
If we divide the period (2.45) into two parts (by 12 numbers in each part)
1
1, 3, 4, 7, 2, 9, 2, 2, 4, 6, 1, 7
2
8, 6, 5, 2, 7, 9, 7, 7, 5, 3, 8, 2
and then sum them in pairs, we find the following regularity (see Table 2.9):
That is, the sums of all 11 mutual numerological values of the period (2.45)
are equal to 9 and only one to the sum 9+9=18; however, its numerological
value is equal to 9. If we use the same reasoning as for Fibonacci numbers, we
can prove the following theorem.
Theorem 2.2. The Lucas numerological series has the period (2.45) of length
24; here the numerological value of the sum of the first 24 Lucas numbers and
all the next sequential 24 Lucas numbers are equal to 9.
Chapter 2
Fibonacci and Lucas Numbers
83
This means that the number 9 is the “numerological
essence” of the Lucas series, that is, it expresses some “sa
cred” property of Lucas numbers.
It is easy to prove that the generalized Fibonacci
numbers given by (2.32) has similar general property (see
Theorem 2.3).
Theorem 2.3. The numerological series of the gen
eralized Fibonacci numbers given by the recursive rela
tion (2.32) has the period of length 24; here the numer
ological value of the sum of the first 24 generalized Fi
bonacci numbers and all the next sequential 24 general
ized Fibonacci numbers are equal to 9.
Table 2.9. Regularity
of the Lucas numero
logical series
1 + 8 = 9
3 + 6 = 9
4 + 5 = 9
7 + 2 = 9
2 + 7 = 9
9 + 9 = 18
2 + 7 = 9
2 + 7 = 9
4 + 5 = 9
6 + 3 = 9
1 + 8 = 9
7 + 2 = 9
2.6. Cassini Formula
2.6.1. Great Astronomer Giovanni Domenico Cassini
Cassini is the name of a famous dynasty of French as
tronomers. Giovanni Domenico Cassini (16251712) is the
most famous of them and the founder of this dynasty. The
following facts illustrate his outstanding contribution to as
tronomy. The name Cassini was given to numerous astro
nomical objects, the “Cassini Crater” on the Moon, the
“Cassini Crater” on Mars, “Cassini Slot” in Saturn’s ring,
and “Cassini Laws” of the Moon’s movement.
However, the name of Cassini is widely known not only Giovanni Domenico
in astronomy, but also in mathematics. Cassini developed a Cassini (16251712)
theory of the remarkable geometrical figures known under the name of Cassini Ovals.
The mathematical identity that connects three adjacent Fibonacci numbers is well
known under the name Cassini Formula. Below we will review this famous formula.
Giovanni Cassini was born on June, 8, 1625 in the Italian town of Perinal
do. He got his education in Jesuit collegiums in Genoa. During 16441650 he
worked in the observatory located near Bologna.
In 1650 Cassini took a professorial chair in mathematics and astronomy at
the University of Bologna. Cassini’s basic scientific works concerned obser
vational astronomy and he became famous, first of all, as a talented observer.
Working in Bologna, for the first time in history, he executed numerous posi
Alexey Stakhov
THE MATHEMATICS OF HARMONY
84
tional observations of the Sun with meridian tools. On the basis of these ob
servations he made new Solar tables published in 1662. Owing to Cassini’s
researches, the Parisian Meridian was established. Cassini’s work provided the
possibility for the creation of the wellknown map of France, the Cassini Map.
The glory of Cassini as an astronomer was so great that in 1669 he was
elected a member of the Parisian Academy of Sciences. According to Pikar’s
recommendation, King Louis XIV invited Cassini to take a position as Direc
tor of Parisian observatory. France became his second native land up to the
end of his life. In Paris, Cassini made a number of outstanding astronomical
discoveries. During 16711684 he discovered several satellites of Saturn. In
1675 he found that the ring of Saturn consists of two parts divided by dark
strip called the Cassini Slot. During 16711679 he observed details of the Lu
nar surface and in 1679 he made a finer map of the Moon. In 1693 Cassini
formulated three empirical laws for the Moon’s movement called Cassini Laws.
Cassini died in Paris on September 14, 1712 at the age of 87, absolutely
blind but a highly honored man.
2.6.2. Cassini Formula for Fibonacci Numbers
The history of science is silent as to why Cassini took such a great interest
in Fibonacci numbers. Most likely it was simply a hobby of the Great astron
omer. At that time many serious scientists took a great interest in Fibonacci
numbers and the golden mean. These mathematical objects were also a hobby
of Cassini’s contemporary, Kepler.
Now, let us consider the Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, 34, … . Take
the Fibonacci number 5 and square it, that is, 52=25. Multiply two Fibonacci
numbers 3 and 8 that encircle the Fibonacci number 5, so we get 3×8=24.
Then we can write:
523×8=1.
Note that the difference is equal to (+1).
Now, we follow the same process with the next Fibonacci number 8, that
is, at first, we square it 82=64 and then multiply two Fibonacci numbers 5 and
13 that encircle the Fibonacci number 8, 5×13=65. After a comparison of the
result 5×13=65 with the square 82=64 we can write:
825×13=1.
Note that the difference is equal to (1).
Further we have:
2
13 8×21=1, 21213×34=1,
and so on.
Chapter 2
Fibonacci and Lucas Numbers
85
We see that the square of any Fibonacci number Fn always differs from the
product of two adjacent Fibonacci numbers Fn1 and Fn+1, which encircle it, by
1. However, the sign of 1 depends on the index n of the Fibonacci number Fn.
If the index n is even, then the number 1 is taken with minus, and if odd, with
plus. The indicated property of Fibonacci numbers can be expressed by the
following mathematical formula:
(2.46)
Fn2 − Fn −1 Fn +1 = (−1)n +1 .
This wonderful formula evokes a reverent thrill, if one recognizes that this
formula is valid for any value of n (we remember that n can be some integer in
limits from ∞ up to +∞). The alternation of +1 and 1 in the expression (2.45)
at the successive passing of all Fibonacci numbers produces genuine aesthetic
enjoyment and a feeling of rhythm and harmony.
2.7. Pythagorean Triangles and Fibonacci Numbers
2.7.1. Pythagorean Theorem
The Pythagorean Theorem is probably the bestknown theorem in all
of geometry. It is remembered by anyone who studied geometry in second
ary school, though having “absolutely forgotten” all mathematics. The es
sence of this theorem is extremely simple. The Pythagorean Theorem as
serts that the sides a, b and c of a right triangle are connected by the fol
lowing formula:
(2.47)
a2+b2=c2.
Despite its ultimate simplicity, the Pythagorean Theorem, in the opinion
of many mathematicians, refers to a category of the most outstanding mathe
matical theorems. Earlier we mentioned Kepler’s widely known statement con
cerning the Pythagorean Theorem and the golden mean. Among all the vast
sea of geometrical results and theorems, Kepler chose only two results that he
named “treasures of geometry”: the Pythagorean Theorem and the “division
in extreme and mean ratio” (the golden mean).
2.7.2. Pythagorean Triangles
Amongst the infinite number of rightangled triangles, it is the Pythagorean
Triangles, for which the numbers a, b, c in (2.47) are integers, which have al
ways caused a special interest.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
86
Pythagorean triangles also can be referred to as
a category of the “treasures of geometry,” and a
study of such triangles is one of the most fasci
nating pages in the history of mathematics.
5
The right triangle with sides 3, 4 and 5 is
the most widely known Pythagorean trian
3
gle (Fig. 2.3). It is called the Sacred or Egyp
tian Triangle because it was widely used in
Ancient Egyptian culture. We mentioned above
4
that this triangle is the main geometrical idea of
the Chephren Pyramid in Giza.
For the Egyptian Triangle in Fig. 2.3 the Py Figure 2.3. Sacred or
thagorean Theorem (2.47) takes the following form:
Egyptian Triangle
(2.48)
42+32=52.
There is a legend that the identity (2.48) was used by the Egyptian land sur
veyors and builders for the definition of a right angle on the earth’s surface. For
this purpose, they used a rope of the length 12m, for example (12=3+4+5). The
rope was divided by means of special loops into three parts of the lengths 3, 4, 5m,
respectively. For the definition of a right angle, the Egyptian land surveyor pulled
one of the parts of the rope, for example, of the length 3m, and then fixed it on the
ground by using special pegs, inserted in two loops. Then, the rope was pulled by
means of the third loop and this loop was fixed by using the third peg. Clearly, the
angle that is formed between the two smaller sides of the triangle accurately equals
90°. Tradition has it that at the foundation of the Pyramid a ritual procedure to
define the right angles at the base of the Pyramid was carried out by the Pharaoh.
2.7.3. FibonacciPythagorean Triangles
There is a question: are there other Pythagorean triangles besides the Egyp
tian Triangle? Rasko Jovanovich [139] gives an original answer to this question.
First, we try to express the identity (2.47) by Fibonacci numbers begin
ning with the first four Fibonacci numbers:
(2.49)
F1 = 1, F2 = 1, F3 = 2, F4 = 3.
Perform the following procedure:
1. Multiply the two middle Fibonacci numbers of the series (2.49):
F2×F3=1×2=2.
2. Double this result: 2×(F2×F3)=2×2=4. The product 4 is the length of
the side a of the Egyptian Triangle in Fig. 2.3, that is, a=2×(F2×F3)=4.
3. Multiply the two external numbers of the series (2.49): F1×F4=1×3=3.
Chapter 2
Fibonacci and Lucas Numbers
87
The product 3 is the length of the side b of the Egyptian Triangle in
Fig. 2.3, that is, b= F1×F4=3.
4. The third, the longest side c of the Egyptian Triangle, is determined if
we add the squares of the interior numbers of the series (2.49), that is, the
numbers (F2)2=12=1 and (F3)2=22=4; then their sum is equal to 1+4=5.
Hence, c=(F2)2+(F3)2=5.
Therefore, we can represent a fundamental identity for the Egyptian Tri
angle (2.48) by Fibonacci numbers as follows:
2
2
2
2
2
2 ( F2 × F3 )  + ( F1 × F4 ) = ( F2 ) + ( F3 )  .


(2.50)
Is this result (2.50) a mere coincidence? To answer this question, we ex
amine the next four Fibonacci numbers:
(2.51)
F2 = 1, F3 = 2, F4 = 3, F5 = 5.
If we repeat the above procedure for the series (2.51), we obtain:
1. 2×3=6
2. 2×6=12; hence, a=12
3. 1×5=5; hence, b=5
4. 22=4, 32=9, 4+9=13; hence, c=13
It is easy to be convinced, that the sides a=12, b=5 and c=13 make up a
Pythagorean triangle because:
122+52=132.
These examples allow us to formulate a general rule to find Pythagorean
triangles by using four adjacent Fibonacci numbers:
Fn , Fn +1 , Fn + 2 , Fn + 3 .
(2.52)
Perform the following:
1. Multiply two middle Fibonacci numbers of the series (2.52): Fn +1 × Fn + 2 .
2. Double this result: 2 × ( Fn +1 × Fn +2 ) ; hence, a = 2 × ( Fn +1 × Fn +2 ) .
3. Multiply the two external Fibonacci numbers of the series (2.52):
Fn × Fn +3 ; hence, b = Fn × Fn +3 .
4. Square the two interior Fibonacci numbers of the series (2.52),
2
2
2
2
( Fn+1 ) and ( Fn+2 ) , and then add them: c + ( Fn +1 ) + ( Fn +2 ) .
For the general case of (2.52) the main identity for Pythagorean triangles
is as follows:
( 2 × Fn +1 × Fn +2 )2 + ( Fn × Fn +3 )2 = ( Fn2+1 + Fn2+2 ) .
2
(2.53)
Now let us consider the Pythagorean triangle for the case n=3. For this
case, the four adjacent Fibonacci numbers are the following:
2, 3, 5, 8.
(2.54)
THE MATHEMATICS OF HARMONY
Alexey Stakhov
88
Then, according to the above algorithm, the sides of the Pythagorean tri
angle can be found in the following manner:
a = 2 × 3 × 5 = 30; b = 2 × 8 = 16; c = 32 + 52 = 9 + 25 = 34.
The Pythagorean Theorem for this case is: 302+162=342.
Finally, for the case n=4 the four adjacent Fibonacci numbers are the follow
ing: 3, 5, 8, 13, and the sides of Pythagorean triangle are equal to, respectively,
a = 2 × 5 × 8 = 80; b = 3 × 13 = 39; c = 52 + 82 = 35 + 64 = 89.
The Pythagorean Theorem for this case is as follows: 802+392=892.
Table 2.10 represents Pythagorean triangles for the initial values n.
It is essential to
Table 2.10. FibonacciPythagorean triangles
note that side c of the
Pythagorean triangles
n
Fn
Fn+1
Fn+2
Fn+3
a
b
c
of Table 2.10 are calcu
1
1
1
2
3
4
3
5
lated by the formula:
2
1
2
3
5
12
5
13
3
2
3
5
8
30
16
34
4
3
5
8
13
80
39
89
6
8
13
21
34
546
272
610
7
13
21
34
55
1428
715
1597
c = Fn2+1 + Fn2+ 2 . (2.55)
By using the identi
ty (2.29), we can write:
c = F2( n +1) +1 ,
that is, the hypotenuse
c of the FibonacciPy
9
34
55
89
144 9790 4869 10946
thagorean triangle is
always equal to some Fibonacci number that is confirmed by Table 2.10.
8
21
34
55
89
3740
1869
4181
2.7.4. LucasPythagorean Triangles
The above procedure for the FibonacciPythagorean triangles also proves
to be valid for Lucas numbers. For example, the first four adjacent Lucas num
bers 1, 3, 4, 7 result in the LucasPythagorean triangle with the sides:
a = 2 × 3 × 4 = 24; b = 1 × 7 = 7; c = 32 + 42 = 9 + 16 = 25.
For this case the Pythagorean Theorem is as follows:
2
24 +72=252.
The next four adjacent Lucas numbers 3, 4, 7, 11 result in the next Lucas
Pythagorean triangle with the sides:
a = 2 × 4 × 7 = 56; b = 3 × 11 = 33; c = 42 + 72 = 16 + 49 = 65.
For this case the Pythagorean Theorem is as follows:
2
56 +332=652.
Table 2.11 represents the LucasPythagorean triangles for the initial
values of n.
Chapter 2
Fibonacci and Lucas Numbers
89
Table 2.11. LucasPythagorean triangles
n
Ln
Ln+1
Ln+2
Ln+3
a
b
c
1
1
3
4
7
24
7
25
2
3
4
7
11
56
33
65
3
4
7
11
18
154
72
170
4
7
11
18
29
396
203
445
5
11
18
29
47
1044
517
1165
6
18
29
47
76
2726
1368
3050
7
29
47
76
123
7144
3567
7985
8
9
47
76
76
123
123
199
199
322
18696
48954
9353
24472
20905
54730
2.7.5. Fibonacci and Lucas Right Triangles
Now, let us consider the identity Fn2 + Fn2+1 = F2n +1 (2.29) as an expression
of the Pythagorean Theorem (2.47) for the right triangle with the sides
a = Fn ; b = Fn +1 and c = F2n +1 . We name Table 2.12. Fibonacci right triangles
these right triangles Fibonacci Right Trian
2
F2 n 1
n
F
Fn 1
gles because they are based on the impor
1
1
1
2
tant identity (2.29) that connects the adja
2
1
4
5
cent Fibonacci numbers. Note that Fibonacci
3
4
9
13
Right Triangles are not Pythagorean. Nev
4
9
25
34
ertheless, they are a very interesting class
5
25
64
89
6
64
169
175
of right triangles. Fibonacci right triangles
7
169
441
610
are given by Table 2.12.
8
441
1156
1597
Note that similar triangles can also be
9
1156
3025
4181
constructed on the basis of Lucas numbers
if we use the identity (2.36) that gives Lucas right triangle with the sides:
2
n
Table 2.13. Lucas right triangles
n
Ln
2
Ln 1
2
5F2 n 1
1
2
3
4
5
6
7
8
9
1
9
16
49
121
324
841
2209
5776
9
16
49
121
324
841
2209
5776
15129
10
25
65
170
445
1165
3050
7985
7305
a = Ln , b = Ln +1 and c = 5F2n +1 .
Table 2.13 gives the Lucas right tri
angles.
The above connection of Fibonacci and
Lucas numbers with Pythagorean triangles
allows us to propose the existence of an in
finite set of Fibonacci and Lucas Pythagore
an triangles. This fact is additional testimo
ny for the fundamental character of Fi
bonacci and Lucas numbers!
Alexey Stakhov
THE MATHEMATICS OF HARMONY
90
2.8. Binet Formulas
2.8.1. Jacques Philippe Marie Binet
In the 19th century, the interest in Fibonacci numbers and the golden
mean in science and mathematics again arose. The mathematical works of the
French mathematicians Lucas and Binet were a bright reflection of this inter
est. We have mentioned above about the French 19th century mathematician
Lucas (18421891), who revived the interest in Fibonacci numbers of the sci
entific 19th century community.
The French mathematician Jacques Philippe Marie
Binet was the other 19th century enthusiast of Fi
bonacci numbers and the golden mean. He was born
on February 2, 1776 in Renje and died on May 12, 1856
in Paris. Following his graduation from the Polytech
nic School in Paris in 1806 Binet worked at the Bridge
and Road Department of the French government. He
became a teacher at the Polytechnic school in 1807
and then Assistant Professor of applied analysis and
descriptive geometry. Binet studied the fundamentals Jacques Philippe Marie
Binet (17761856)
of matrix theory and his work in this direction was
continued later by other researchers. He discovered in 1812 the rule for ma
trix multiplication, which glorified his name more than any of his other works.
Binet worked in other areas in addition to mathematics. He published
many articles on mechanics, mathematics and astronomy. In mathematics,
Binet introduced the notion of the “beta function”; also he considered the
linear difference equations with alternating coefficients and established some
metric properties of conjugate diameters, etc. Among his many honors, Binet
was elected to the Parisian Academy of sciences in 1843.
Binet entered Fibonacci number theory as author of the famous mathe
matical formulas called Binet Formulas. These formulas link the Fibonacci and
Lucas numbers with the golden mean and, undoubtedly, belong amongst his
tory’s most famous mathematical formulas.
2.8.2. Deducing Binet Formulas
In order to deduce Binet formulas, we must first consider the remarkable
identity that connects the adjacent degrees of the golden mean:
Chapter 2
Fibonacci and Lucas Numbers
91
(2.56)
τn = τn −1 + τn −2 ,
where τ is the golden mean and n takes its values from the set: 0, ±1, ±2, ±3, ….
Now, let us write the expressions for the zero, first and minusfirst terms of
the golden mean:
2 + 0× 5 1 1+ 5
−1 + 5
.
, τ =
, and τ −1 =
(2.57)
2
2
2
Let us remember that the degrees of τ are connected one to the other by
the following identity:
(2.58)
τ2 = τ + 1,
that is a partial case of the general identity (2.56) for the case n=2.
By employing the expressions of (2.57) and the identities (2.56) and (2.58),
we can represent the second, third and fourth degrees of the golden mean as
follows:
3+ 5 3
4+2 5 4
7+3 5
; τ = τ 2 + τ1 =
; τ = τ3 + τ2 =
.
τ 2 = τ1 + τ0 =
(2.59)
2
2
2
Is it possible to see some regularity in the formulas (2.57) and (2.59)?
First of all, we can see that each expression for any degree of the golden mean
has the following typical form:
τ0 = 1 =
A+ B 5
.
2
What are the numerical sequences A and B in these formulas? It is easy to
see that the series of the numbers А is the number sequence 2, 1, 3, 4, 7, 11, 18, …,
and the series of the numbers В is the number sequence 0, 1, 1, 2, 3, 3, 5, 8, ….
Thus, the first sequence is Lucas numbers Ln and the second one is Fibonacci
numbers Fn! It follows from this reasoning that the general formula that allows
representation of the nth degree of the golden mean by Fibonacci and Lucas
numbers has the following form:
L + Fn 5
τn = n
.
(2.60)
2
Note that the formula (2.60) is valid for each integer n taking its values
from the set 0, ±1, ±2, ±3, ….
By using the formula (2.60), it is possible to represent the “extended” Fi
bonacci and Lucas numbers by the golden mean. For this purpose, it is enough
to write the formulas for the sum or difference of the nth degrees of the gold
n
−n
en mean τn + τ − n and τ − τ as follows:
(L + L− n ) + (Fn + F− n ) 5
τn + τ − n = n
(2.61)
2
Alexey Stakhov
THE MATHEMATICS OF HARMONY
92
( Ln − L− n ) + ( Fn − F− n ) 5
.
(2.62)
2
Now, let us consider the expressions of the formulas (2.61) and (2.62) for
the even values of the index n=2k. For this purpose, recall once again the fol
lowing wonderful property of Fibonacci numbers: for the even values of n the
Fibonacci numbers F2k and F2k are equal by absolute value and opposite by sign,
that is, F2k=F2k and likewise for the Lucas numbers L2k and L2k, that is,
L2k=L2k. Then for the case of n=2k the (2.61) and (2.62) take the following form:
τn − τ − n =
τ2 k + τ −2 k = L2 k
(2.63)
(2.64)
τ2 k − τ −2 k = F2 k 5 .
For the odd n=2k+1 we have the following relations for the “extended”
Fibonacci and Lucas numbers: F2k1=F2k+1 and L2k1=L2k+1. Then, for this case
the formulas (2.63) and (2.64) take the following form:
τ2 k +1 + τ (
− 2 k +1)
= F2 k 5
(2.65)
− 2 k +1
(2.66)
τ2 k +1 − τ ( ) = L2 k +1 .
We can now represent the formulas (2.65) and (2.66) in the following com
pact forms:
 τn + τ− n for n = 2k
Ln =  n
−n
 τ − τ for n = 2k + 1
(2.67)
 τn + τ− n
for n = 2k + 1

5

.
Fn =  n
−n
 τ − τ for n = 2k

5
(2.68)
The analysis of the formulas (2.67) and (2.68) provide us with “aesthetic
pleasure” and once again we are convinced of the power of the human mind! We
know that the Fibonacci and Lucas numbers are always integers. On the other
hand, any degree of the golden mean is an irrational number. It follows from this
that the integer numbers Ln and Fn can be represented with the help of the for
mulas (2.67) and (2.68) by the special irrational number, the golden mean!
For example, according to (2.67) and (2.68) we can represent the Lucas
number 3(n=2) and the Fibonacci number 5(n=5) as follows:
2
−2
1+ 5  1+ 5 
+
3=
 2   2  ,

 

(2.69)
Chapter 2
Fibonacci and Lucas Numbers
93
5
−5
1+ 5  1+ 5 

 +

2
2  .



(2.70)
5=
5
It is easily proven that the identity (2.69) is valid because according to
(2.60) we have:
2
−2
1+ 5 
 1+ 5 
L2 + F2 5 3 + 5
L−2 + F−2 5 3 − 5
=
=
 =
 =

and 
.
2
2
2
2
 2 
 2 
If now we substitute these expressions to the right of the formula (2.69),
then the righthand side of (2.69) can be represented as follows:
3+ 5 3− 5
+
.
(2.71)
2
2
By adding the items of (2.71), we can see that all “irrationalities” in (2.71)
are mutually eliminated and we obtain the number 3 from the sum, that is, the
identity (2.69) is valid.
We can be convinced in the validity of the identity (2.70), if we remember
that according to (2.60) we have the following representations:
5
1+ 5 
L5 + F5 5 11 + 5 5
=
 =

2
2
 2 
−5
1+ 5 
L−5 + F−5 5 −11 + 5 5
=
 =
and 
.
2
2
 2 
If we substitute these expressions into (2.70), then we obtain the follow
ing expression for the righthand side of (2.70):
10 5
= 5,
2 5
whence follows the validity of the identity (2.70).
Note that this reasoning has a general character, that is, for any Lucas or
Fibonacci numbers that are given by formulas (2.67) and (2.68), all “irratio
nals” in the righthand parts of (2.67) and (2.68) are always mutually elimi
nated and we obtain integers as the outcome!
2.8.3. A Historical Analogy
The situation of the mutual elimination of all “irrationals” in the formulas
(2.67) and (2.68) reminds one of the situations that appeared in mathematics
with the introduction of complex numbers. In the 16th century, Italian mathe
maticians contributed significantly to the development of algebra: they solved
THE MATHEMATICS OF HARMONY
Alexey Stakhov
94
radicals with equations of the 3rd and 4th degrees. Cardano’s famous book, The
Great Art (1545), contains the algebraic solution of the cubic equation:
(2.72)
x 3 + px + q = 0
according to the formula x = u + v, where
2
3
2
3
q
q
q
p
q  p
p
+   +   ; v = 3 − −   +   ; uv = − .
2
2
2
3
3
2  3
   
It was proven algebraically that there are three roots of the algebraic equa
tion (2.72), namely:
2
3
q  p
1. For the case ∆ =   +   > 0 , Eq. (2.72) has one real root and
2  3
two complex conjugate roots; for example, the equation x3+15x+124=0 with
∆>0 has the following roots: x1=4; x2,3 = 2 ± 3i 3 .
2. For the case ∆=0, p≠0, q≠0, Eq. (2.72) has three real roots; for exam
ple, the roots of the equation x312x+16=0 are: x1=4, x2,3=2.
3. For ∆<0 we have the most interesting instance, the socalled “non
reducible” case, where we need to extract the root of the 3rd degree from
complex numbers and the cubic roots u and v are complex numbers. Nev
ertheless, in this case the equation (2.72) has real roots. For example, the
equation x321x+20=0 with
u=3 −
(2.73)
∆ = −243, u = 3 −10 + −243 , v = 3 −10 − −243
has real roots 1, 4 and 5 !
This fact seemed paradoxical to the 16th century mathematicians! Real
ly, all factors of the equation x321x+20=0 are real numbers, all its roots are
real numbers, however, the intermediate calculations result in the “imagi
nary,” “false,” “nonexistent” numbers of the kind (2.73). Mathematicians were
in a very difficult situation again (starting with the discovery of irratio
nals). To completely ignore the numbers of the kind (2.73) would mean to
refuse the general formulas for the solution of algebraic equations of the 3rd
degree, and other remarkable mathematical achievements. On the other hand,
to recognize that these obtrusively appearing “monstrous” numbers such as
(2.73) are equivalent with real numbers was inadmissible from the point of
view of common sense. For a long time the “monstrous” numbers of the kind
(2.73) were not recognized by many mathematicians. For example, Descartes
considered that the complex numbers do not have any real interpretation
and are doomed forever to remain only “imaginary” numbers (the name
“imaginary numbers” came into mathematics in 17th century after Des
cartes). Many eminent mathematicians, in particular, Newton and Leibniz,
adhered to the same opinion.
Chapter 2
Fibonacci and Lucas Numbers
95
17th Century English mathematician Vallis in the book Algebra: Histori
cal and Practical Treatise (1685) pointed out a possible geometric interpreta
tion of complex numbers. Ultimately, the complex numbers came into use in
18th century after the works of the French mathematician Moivre (16671754)
who introduced the following wellknown formula:
( cos ϕ ± i sin ϕ ) n = cos nϕ ± sin nϕ.
(2.74)
After the introduction of Moivre’s formula (2.74), a representation of the
complex numbers in trigonometric form came into use, which facilitated a
solution of numerious mathematical problems. However, the famous Euler’s
Formulas became the “moment of celebration” for complex numbers. By using
Moivre’s formula (2.74), Euler proved the following formulas for trigonomet
ric functions:
e xi + e − xi
e xi − e − xi
, sin x =
.
(2.75)
2
2i
Note that finding the connection between trigonometric and exponential
functions expressed by Euler’s formulas emphasize a fundamental connection
between the numbers π and е, two numerical constants of mathematics, that
play an important role in mathematics, in particular, in the development of
the complex number concept.
By returning back to Binet formulas (2.67) and (2.68) and taking into
consideration our reasoning concerning the complex numbers, we may sup
pose that Binet formulas touch upon some rather deep numbertheoretical
problems that are at the intersection of integers (Fibonacci and Lucas num
bers) and irrationals (the golden mean). Further, in Chapter 9 we will try to
broaden this idea by considering the number systems with irrational bases,
which may overturn our ideas about number systems.
cos x =
2.9. Fibonacci Rectangle and Fibonacci Spiral
2.9.1. Fibonacci Rectangles
Let us consider the Fibonacci series: 1, 1, 2, 3, 5, 8, 13, 21, …. Take two
squares with all of the sides equal to 1 (the area of each square is equal to 1)
and put them together. As a result we get the 2×1 or “double square” rectan
gle. Then, construct a new square of the size 2×2 on the longer side of the
“double square.” Here we obtain a new rectangle of size 3×2. Then, construct
Alexey Stakhov
THE MATHEMATICS OF HARMONY
96
a new square of size 3×3 on the longer side of the preceding rectangle; as a
result, we obtain a new rectangle by the size 5×3. Continuing this process,
results in rectangles, with side equal to adjacent Fibonacci numbers, that is,
the rectangles with the following sizes: 8×5, 13×8, 21×13, and so on (Fig. 2.4).
5
5
8
1 1 1 1 1
2
3
1 1
2
3
1 1
2
3
1 1
2
Figure 2.4. Fibonacci rectangles
Such rectangles are called Fibonacci Rectangles. Consider a sequence of
the ratios of the Fibonacci rectangles sides: 1:1, 2:1, 3:2, 5:3, 8:5, 13:8, 21:13,
34:21, …. As these ratios aim for the golden mean, we can conclude that the
Fibonacci rectangles aim for the golden rectangle, that is, the Fibonacci rect
angles are sequential approximations of the golden rectangle.
2.9.2. Fibonacci Spiral
Fibonacci rectangles in Fig.
2.4 consist of the squares with
sides 1, 2, 3, 5, 8, 13, …. Now, in
each square we draw an arc that
34
is equal to a quarter of a circle as
55
shown in Fig. 2.5. If we connect
5
8
these arcs, we obtain a curve that
21
is suggestive of a spiral by its form
13
(Fig. 2.5). Strictly speaking, this
curve is not a spiral from the
Figure 2.5. Fibonacci spiral
mathematical point of view. How
ever, it is a very good approximation of the golden spirals that are widely met
in nature. The curve in Fig. 2.5 we call a Fibonacci Spiral.
2.9.3. Fibonacci Spirals in Nature
The great poet and scientist Goethe considered a spiral form as one of the
characteristic attributes of all living organisms, as a manifestation of the most
secret essence of life. Short moustaches of plants and horns of rams are twisted
by a spiral, a growth of tissues in trunks of trees is carried out by a spiral, sun
Chapter 2
Fibonacci and Lucas Numbers
97
flower seeds are disposed by a spiral, and so on. Everyone admires the forms of
shells that are constructed under the spiral law. Let us begin from the spiral
form of the Nautilus (Fig. 2.6a). If we compare its shape with the Fibonacci
spiral (Fig. 2.5), we can conclude that the nautilus shell is constructed accord
ing to the Fibonacci spiral principle. The motives of the Fibonacci spiral can be
found in the shape of the Galaxy (Fig. 2.6b) and other sea shells (Fig. 2.6c).
a)
b)
Figure 2.6. Fibonacci spirals in Nature
c)
2.10. Chemistry by Fibonacci
2.10.1. Law of Multiple Ratios
There is an opinion that the accumulated sum of knowledge in some disci
pline can be called science only when it passes into a precise, quantitative
analysis beyond the merely qualitative perspective. To the end of the 18th
century, chemistry accumulated a significant volume of knowledge and chem
ists had learned to decompose many complex substances into simple ones and
to build the complex substances from the simple ones. During this process the
problem of quantitatively describing various chemical compounds composing
complex compounds appeared. It was necessary for the creation of a graceful
theory of chemical structures, which would meet the requirements for the pro
duction practice of various chemical products.
The Law of Constant Proportions of Chemical Compounds is one of the funda
mental chemical laws. This law came into chemical science after research by French
scientist Prust (17541826). By studying chemical compounds, in particular the
oxides of metals, he came to the conclusion that the chemical compounds have a
strictly constant structure that is not dependent on the conditions of their forma
Alexey Stakhov
THE MATHEMATICS OF HARMONY
98
tion. Thanks to the works of the English scientist Dalton (17661844), the atom
ic doctrine came into chemistry and the Law of Simple Multiple Ratios was formu
lated. According to this law, simple integervalued ratios between atoms in chem
ical structures exist. Now students know that the structure of water is described
by the formula H2O, of common salt NaCl, and zinc oxide ZnO. Chemistry became
a precise science. A new branch of chemistry arose that studies the ratios of atoms
in chemical structures called Stoichiometry.
The discovery of the law of multiple ratios is one of the remarkable achieve
ments of chemistry as a science: a beautiful simplyordered system of chemical
compounds appeared from the chaos of atomic representations. Atoms of differ
ent elements can form different combinations that are connected by the forces of
chemical connection. However, only some of them are stable; the other chemical
combinations perish by disintegrating into more stable compounds. Only those
combinations of atoms of different elements will be steady, if they correspond to
simple integervalued ratios of their components. This idea is surprisingly simple
and clear and corresponds completely to the Pythagorean doctrine about the dom
inating role of integer numbers in the organization of the Universe!
However, such formulation of the main chemical law evokes some bewil
derment. It is not clear as to what the “simple integervalued ratios” of atoms in
the formulas of chemical compounds really means. While we studied rather sim
ple chemical compounds, atomic ratios in them usually corresponded to “small”
numbers, for example, H2O, Al2O3, Fe3O4, As2O5. However, the range of the stud
ied chemical compounds started to broaden rapidly. Formulas of chemical com
pounds appeared there with the stoichiometric factors of 7, 9, 15, 21, etc. When
the chemists started studying the structure of the organic compounds, it be
came inconvenient to speak about the “simple integervalued ratios.” The chem
ical structure of the bacteriophage is a peculiar champion in stoichiometry be
cause it has the following formula: C5750H7227N2215O4131S590. What do the ratios of
the “small” integers mean in this formula? Here we see fourdigit numbers!
Thus, not all is so simple in stoichiometric laws: simplicity here is com
bined with complexity, and the question about the possible ratios of atoms in
compounds remains open.
2.10.2. Research by the Ukrainian Scientist Nikolai Vasyutinsky
We will not delve too deeply into the chemistry of different compounds.
We are interested in only one problem — whether Fibonacci numbers are found
in the formulas of chemical compounds. The Ukrainian chemist Nikolai
Vasyutinsky attempted to give the answer to this question in his book [31].
Chapter 2
Fibonacci and Lucas Numbers
99
By analyzing oxides of uranium and chromium, Vasyutinsky found that
these chemical compounds are based on Fibonacci numbers. With the oxides
of uranium, the structure of the generating oxides is changing not continu
ously, but spasmodically, from one steady compound with a certain integer
valued ratio of atoms into another one. Between the uranium oxides UO2 and
UO3 a lot of the intermediate compounds are forming; their structures are
described by the formulas U2O5, U3O8, U5O13, U8O21, U13O34. We can see that
the atomic ratios in these compounds are equal to the ratios of Fibonacci num
bers: 2:5, 3:8, 5:13, 8:21, 13:34. It is easy to prove that such ratios aim in the
limit for the quadrate of the golden mean! Each of the described uranium ox
ides can be represented as the sum of two oxides UO2 and UO3 taken in the
different proportions, for example: U5O13=3UO3 +2UO2, U8O21=5UO3+3UO2.
Here the factors of the oxides UO2 and UO3 correspond to the adjacent Fi
bonacci numbers! This means that the structures of the aboveconsidered ura
nium oxides are subordinated completely to the Fibonacci number regularity.
Note that according to Vasyutinsky’s research the chromium oxides Cr2O5,
Cr3O8, Cr5O13 have structures that are described by Fibonacci numbers.
By considering the equalities of the type U5O13=3UO3 +2UO2, we can find
their similarity to the algebraic golden mean equation of the 4th degree
x4=3x+2 that is used for the description of the butadiene structure. By com
paring the equality U8O21=5UO3+3UO2 with the algebraic golden mean equa
tion of the 5th degree x5=5x+3, we can see that they also have a similar math
ematical structure. Perhaps these analogies can be the beginning of rather
interesting research into the field of stoichiometry.
It is generally accepted procedure to determine the structures of the
chemical compounds by the ratio of atoms of the elements that is included
in this compound. However, it is possible to estimate the chemical com
pounds by the ratios of the atoms (ions) of different elements to the mo
bile valence electrons that are responsible for the formation of chemical
connections between atoms. So, for example, in the chromium oxide Cr 2O5
the 10 valence electrons correspond to the 7 atoms of the chromium and
oxygen. Making similar calculations for all the above oxides, we obtain
the following ratios of the sums of the atoms to the sums of the valence
electrons: 10:7, 16:11, 26:18, 42:29, 68:47. Note that the numerators of these
fractions are double Fibonacci numbers (10=2×5, 16=2×8, 42=2×21,
68=2×34) and the denominators are the Lucas numbers 7, 11, 18, 29, 47. If
we decreased sequentially the numerators and the denominators of these
fractions on the Fibonacci numbers that correspond to the metal atom
quantity in compounds, that is, on 2, 3, 5, 8, 13, we obtain the series of
Alexey Stakhov
THE MATHEMATICS OF HARMONY
100
ratios of adjacent Fibonacci numbers: 8:5, 13:8, 21:13, 34:21, 55:34; in the
limit these ratios aim for the golden mean.
Thus, the Ukrainian chemist Vasyutinsky convincingly demonstrated that
chemical compounds organized by Fibonacci Numbers do exist.
2.11. Symmetry of Nature and the Nature of Symmetry
2.11.1. Basic Concepts of Symmetry
Symmetry is one of the most fundamental scientific concepts that among
the concept of Harmony has a relation to practically all branches of Nature,
Science and Art. The outstanding mathematician Hermann Weil evaluated
the role of symmetry in modern science in the following words:
“Symmetry, as though wide or narrow we did not understand this word,
there is an idea, with the help of which a person attempts to explain and cre
ate an order, beauty and perfection.”
What is symmetry? When we look at a mirror we can see us in its reflec
tion; this is an example of mirror symmetry. The mirror reflection is an exam
ple of the socalled Orthogonal Transformation that changes an orientation. In
the general case Symmetry in mathematics is perceived as a transformation of
space (plane), when each point M of space (plane) turns into the other point
M′ with respect to some plane (or a straight line) a; here, the line segment
MM is perpendicular to a plane (or a straight line) a and is divided by it in
half. The plane (or the straight line) a is called Plane (or Axis) of Symmetry.
Plane of Symmetry, Symmetry Axis and Center of Symmetry are fundamental
concepts of symmetry. A plane of symmetry P is a plane that divides the figure into
two mirrorsymmetrical parts that are disposed one to another as some subject and
its mirror reflection. For example, the isosceles triangle ABC shown in Fig. 2.7 at
B
p
A
D
C
Figure 2.7. A symmetry of triangle and rectangular parallelepiped
Chapter 2
Fibonacci and Lucas Numbers
101
the left is divided by the altitude BD into two mirrorsymmetrical halves ABD and
BCD; thus, the altitude BD is the “track” of the plane of symmetry P that is perpen
dicular to the plane of the triangle. We say that the isosceles triangle ABC has only
the plane of symmetry P. In Fig. 2.7 at the right a rectangular parallelepiped (match
box) is shown; it has three orthogonal planes of symmetry 3P. It is easy to show, that
a cube has nine planes of symmetry — 9P. These examples could be continued.
A symmetry axis L is called a straight line, around which a symmetrical figure
can be turned several times in such a manner that each time the figure coincides
with itself in space. For example, the equilateral triangle has the symmetry axis L3
that is perpendicular to the triangle plane in the center of the triangle. It is clear
that there are three ways of turning a triangle around the symmetry axis L3 when
the triangle coincides with itself. It is clear, that a square has the symmetry axis L4
and the pentagon L5. The cone also has a symmetry axis; there are an infinite
number of the turns of the cone around the symmetry axis. This means that the
cone has the symmetry axis of the type L∞.
Finally, a symmetry center C of a figure is constructed inside the figure
when any straight line, drawn through the point C, meets the identical points
on the figure at equal distances from the center C. A sphere is a perfect exam
ple of a figure with center symmetry.
2.11.2. Symmetry of Crystals
For many centuries, the geometry of crystals seemed a mysterious and un
solvable riddle. In 1619, the Great German mathematician and astronomer Jo
hannes Kepler (15711630) paid close attention to the sixfold symmetry of snow
flakes. He tried to explain it by the fact that their crystals are constructed from
small identical balls that are connected one to another. Kepler’s idea was devel
oped by the English polymath Robert Hooke (16351703) and the Russian poly
math Mikhail Lomonosov (17111765), who made important contributions to
Russian literature, education and science. They also assumed that it is possible to
liken the elementary particles inside crystals to densely packed balls. Presently
the Principle of Dense Spherical Packing underlies structural crystallography, only
the spherical particles of the ancient authors are replaced now by atoms and ions.
Fifty years after Kepler, the Danish geologist and crystallographer Nicolas Stenon
(16381686) for the first time formulated the main idea of crystal growth: “Crystal
growth is implemented not from within, similar to plants, but by means of the su
perposition on the external planes of the crystal’s smallest particles, which are brought
from outside by certain liquids.” This idea about crystal growth as the outcome of
the sediment forming new stratums of substance on crystal faces preserves its sig
Alexey Stakhov
THE MATHEMATICS OF HARMONY
102
nificance until now. It was Nicolas Stenon who discovered the main law of geomet
ric crystallography — the law of Constancy of Interface Angles.
The law of Constancy of Interface Angles became a reliable basis for the de
velopment of geometric crystallography and provided the richest material for ver
ification of the true symmetry of crystallographic structures. The French researcher
Rene Just Hauy assumed that the crystals consist not of the small balls (according
to the assumption of Kepler, Hooke and Lomonosov), but of the molecules of
parallelepiped shape, and these molecules are the extreme small splinters of the
same shape. In other words, the crystals are peculiar “masonries” forming molec
ular “bricks.” Despite all the naivety of this theory from the modern point of view,
this idea did play an important role in the history of crystallography and gave an
impetus to the origin of the theory of crystal lattice structures.
The French crystallographer Brave was Hauy’s direct follower. He replaced
the molecular “bricks” by points, the centers of molecular weight. As a result
of this approach, the spatial lattices of crystals were obtained. By developing
a hypothesis about the lattice structure of all crystals, Brave created the fun
damentals of modern structural crystallography long before the experimental
research of crystal structures with the help of Xrays.
We can see in Fig. 2.8 the spatial crystal’s lattices of the common salt NaCl
(a) and the calcium oxide CaO (b). In all crystals, we can see the set of the
identical atoms located like the points of a spatial lattice. The straight lines, on
which atoms in the lattice are located, are named rows, and the planes, filled by
atoms, are named planar lattices.
A presentation about the lattice
structure of crystals allowed us to give
a general definition of crystals. It is
accepted that crystals are solid bod
ies, which consist of particles (atoms,
ions, molecules), located the strictly
a)
regular knotlike spatial lattice. This
b)
general definition can be applied to Figure 2.8. Spatial crystal lattices of NaCl (а)
and CaO (b)
all types of Nature’s crystals.
However, already in 1830, long before Brave, the German professor Johannes
Fridrich Hessel (17961872) published the long article Crystallometry, where he
developed an approach to crystal geometry based on the symmetry concept. In
this article, Hessel gave a full research report of the set of symmetry elements of
crystals and spatial geometric figures. Only in 1890, that is, in 60 years after pub
lication of this article, and in 18 years after the death of its author, crystallogra
phers could evaluate the significance of Hessel’s outstanding discovery.
Chapter 2
Fibonacci and Lucas Numbers
103
In 1890, the eminent Russian crystallographer Fedorov proved the exist
ence of the 230 sets of symmetry elements for all existing types of crystals.
Fedorov’s symmetry groups corresponding to the geometric laws of atom lo
cation in crystal structure underlies modern structural crystallography.
A presentation about the lattice structure of crystals, in particular, a con
cept of “planar lattice,” on which the knots of the crystal lattice are located,
resulted in the discovery of the Major Law of Crystallographic Symmetry. Ac
cording to this law of crystal, symmetry axes the first, second, third, fourth and
sixth orders are allowed. However, it is impossible for crystals to have the fifth
order symmetry or anything greater than the sixth order of symmetry.
According to the main crystallography law, there is a fundamental difference
between the symmetry of the mineral world and the symmetry of the living world,
where fivefold symmetry is widely used. For crystals, the fivefold axis of sym
metry and the symmetry axes greater than the 6th order are prohibited: this strict
rule of classical crystallography existed until 1982, when the Israeli physicist Dan
Shechtman proclaimed the startling discovery of quasicrystals.
2.11.3. Symmetry Laws in Nature
The concept of symmetry is used widely in
physics. If the laws that determine relations be
tween physical magnitudes and a change of these
magnitudes in the course of time do not vary at
the definite operations (transformations), they
say, that these laws have symmetry (or they are
invariant) with respect to the given transforma
tions. For example, the law of gravitation is val
id for any points of space, that is, this law is in
variant with respect to the system of coordinates.
In the opinion of outstanding Russian sci
entist and academician Vernadsky, “symmetry
encompasses properties of all fields of physics
and chemistry.”
The Pythagoreans paid very close attention
to the phenomenon of symmetry in Nature that
was connected with their Harmony doctrine. The
two kinds of symmetry, Mirror and Radial, are
widespread throughout Nature. A butterfly, a leaf,
and a beetle (Fig. 2.9a) have “mirror” symmetry
a)
b)
Figure 2.9. Natural forms with
“bilateral” (а) and “radial” (b)
symmetries
Alexey Stakhov
THE MATHEMATICS OF HARMONY
104
and often such type of symmetry is called Leaf Symmetry or Bilateral Symmetry. A
mushroom, a chamomile, and a pine tree (Fig. 2.9b) have a “radial” symmetry, and
often such type of symmetry is called a ChamomileMushroom Symmetry.
Already in the 19th century the researchers in this area came to the con
clusion that symmetry of natural forms largely depend upon the influence of
the Earth’s gravitational forces that have the symmetry of a cone in each point.
In the outcome, the following law was found:
“Everything that grows or moves in a vertical direction, that is, upwards or
down relative to the Earth’s surface is subordinated to the “radial” (“chamomile
mushroom”) symmetry. Everything that grows and moves horizontally or with an
inclination relative to the Earth’s surface is subordinated to the “bilateral” or “leaf”
symmetry.”
2.11.4. Symmetry Laws in Art
The principle of symmetry is used widely in Art. The curbs in architectural
and sculptural works, the ornamental designs in the applied art are examples of
the application of the laws of symmetry.
Symmetry together with the golden mean principle is often used in works
of art. Raphael’s picture The Engagement of Virgin Mary (Fig. 2.10) is just
such an example.
Figure 2.10. Raphael’s picture “The Engagement of Virgin Mary” and its harmonic analysis
Chapter 2
Fibonacci and Lucas Numbers
105
2.12. Omnipresent Phyllotaxis
2.12.1. A Helical Symmetry
Everything in Nature appears subordinate to stringent mathematical laws.
For example, leaf disposition on plant stems also has stringent mathematical reg
ularity, a phenomenon called Phyllotaxis in botany. The essence of phyllotaxis
consists of a spiral disposition of
leaves on plant stems, trees, petals
in flower baskets, seeds in a pi
necone and those of a sunflower
head, etc. This phenomenon was
already known to Kepler and was
a subject of discussion of many sci
entists throughout the years, in
cluding Leonardo da Vinci, Turing,
Figure 2.11. A helical symmetry
Veil, and so on. Much more com
posite concepts of symmetry are used in the phenomenon of phyllotaxis, in partic
ular, the concept of Helical Symmetry. Let us examine, for example, a disposition
of leaves on the plant stem (Fig. 2.11). We can see in Fig. 2.11 that the leaves are
on different heights on the stem along the he
lical curve that encircles the stem. To move
up from the lower leaf to the higher one, it is
5
6
necessary virtually to turn the leaf at some
angle around the vertical axis and then to raise
5
the leaf up a definite distance. This transfor
4
mation is the essence of helical symmetry.
4
Let us examine typical helical axes that
can appear along the plant stems (Fig. 2.12).
3
In Fig. 2.12a we can see the stem of a plant
3
with helical symmetry axis of the 3rd order.
We can unite all leaves of the plant by any
2
2
virtual line. We start from leaf 1. To move
1
up from leaf 1 to leaf 2, it is necessary to turn
1
leaf 1 around the stem axis 120° counter
clockwise (if looking from below) and then
(a)
(b)
move up leaf 1 along the stem in the verti
Figure 2.12. The helical axis on
stems of plants
cal direction until it coincides with leaf 2. If
Alexey Stakhov
THE MATHEMATICS OF HARMONY
106
we repeat a similar operation, we can move up from leaf 2 to leaf 3 and then on
to leaf 4. It is necessary to note that leaf 4 is on the same line with respect to
leaf 1 (as though it repeats on the higher level). It is clear if we want to move
up from leaf 1 to leaf 4 we have to make three turns of 120o i.e. we have to
carry out a full rotation around the stem axis (120°×3=360°).
Botanists called the rotation angle of the helical axis the Leaf’s Diver
gence Angle. A vertical straight line that is parallel to the stem of the plant
connecting two leaves is called an Ortholine. The line segment of leaf 1 to leaf
4 of the ortholine corresponds to a full transformation of the helical axis. We
will see further that the number of turns around the stem axis to move up
from the bottom leaf to the next leaf directly above it along the ortholine,
can be equal not only to 1, but also to 2, 3, and so. This number of turns is
called the Leaf’s Cycle. In botany, it is acceptable to characterize helical leaf
location by the ratio m/n where the numerator m is a number of turns in the
leaf’s cycle, and the denominator n is the number of leaves in this cycle. In the
above case, we have a helical axis of 1/3.
In Fig. 2.12b the helical axis of fivefold symmetry with a leaf cycle 2 is
presented; this means that for the transition from leaf 1 to leaf 6 it is necessary
to make two full turns. The fraction 2/5 characterizes the given helical axis;
the leaf divergence angle is equal to 144° that is, we have: 360/5=72°;
72°×2=144°. Note that there are also more complex helical axes, for example,
of the kind 3/8, 5/13, and so forth.
There is a question as to what values can take the numbers m and n that
describe the helical axis of the kind m/n. And here Nature gives us the follow
ing surprise called the Law of Phyllotaxis.
Botanists assert that the fractions that describe the helical axes of plants
build up a strict mathematical sequence of the following kind, for example:
1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34, ….
(2.76)
Note that the fractions in the sequence (2.76) are the ratios of two adja
cent Fibonacci numbers beginning with number one. It is easy to prove, that
the sequence (2.76) aims for the number τ2=0.382, that is, to the inverse of
the square of the golden mean.
Botanists found that different plants have different ratios of the above
sequence (2.76). For example, the fraction 1/2 is peculiar to cereals, birch and
grapes; 1/3 to sedge, tulip, and alder; 2/5 to pear, currant, and plum; 3/8 to
cabbage, radish, and flax; 5/13 to firtree and jasmine, etc.
What is the “physical” cause that underlies the phyllotaxis law (2.76)?
The answer is very simple. It proves to be the disposition of leaves that allow
for the maximum inflow of solar energy to the plant.
Chapter 2
Fibonacci and Lucas Numbers
107
2.12.2. Densely Packed Phyllotaxis Structures
The phyllotaxis phenomenon shows itself in inflorescences and densely
packed botanical structures, such as, pinecones, pineapples, cacti, sunflowers,
cauliflowers and many other structures.
Figure 2.13 gives examples of phyllotaxis objects (cactus, sunflower, cone
flower, Romanescue cauliflower, pineapple, pinecone), in which the phyllo
taxis law is based on the numerical sequence (2.30) that consists of the ratios
of adjacent Fibonacci numbers. This means that the seeds or small parts on
the surface of such botanical objects are located at the crossings of lefthand
and righthand spirals; here the ratio of the numbers of lefthand and right
hand spirals is always equal to the ratio of adjacent Fibonacci numbers (2.30).
The same regularity is observed in baskets of flowers.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 2.13. Phyllotaxis structures: (a) cactus; (b) head of sunflower; (c) coneflower; (d)
Romanescue cauliflower; (e) pineapple; (f) pine cone
Geometric models of phyllotaxis structures in Fig. 2.14 provide a clearer
representation of this unique botanical phenomenon.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
108
(a)
(b)
(c)
Figure 2.14. Geometric models of phyllotaxis structures:
(a) pineapple; (b) pinecone; (c) shasta daisy
Thus, we find strict mathematics in the dispositions of leaves on plant
stalks, flower petals, in the crosssection of an apple (pentacle), in the spiral
dispositions of seeds of a pinecone, pineapple, cactus and the head of a sun
flower. And this mathematical law is expressed by Fibonacci numbers and
therefore the golden mean! Once again, it appears as though Nature is subor
Chapter 2
Fibonacci and Lucas Numbers
109
dinate to a coherent plan, the uniform Law of the Golden Section! To discover
and explain this fundamental law of Nature in all its expressions is one of the
most important problems of science and philosophy.
2.12.3. Use of Phyllotaxis Lattices in Painting
In the above we mentioned a wideranging use of the golden rectangle and
other golden geometrical figures in artwork. One art method used by artists of
the Renaissance was the use of phyllotaxis grids in painting. This is not unlike the
botanical phyllotaxis phenomenon in which Nature appears to design pinecones,
pineapples, cacti, sunflower heads, and many other botanical structures.
According to phyllotaxis law, cactus spines are located along Fibonacci
spirals; here the adjacent Fibonacci numbers 21 and 34 are the numbers of the
lefthand and righthand spirals. If we unroll on a plane the spines of a cactus,
we can obtain a raster grid (Fig. 2.15). In the raster grid in Fig. 2.15 the in
clined lines with righthand and lefthand inclination represent the principle
of disposition of the spines on the cactus surface. This raster grid has 21 lines
with righthand inclination and 34 lines with lefthand inclination. The net of
the lines in Fig. 2.15, Phyllotaxis Raster Grid, from an aesthetic point of view
looks as if it is a golden rectangle. Many artists of the Renaissance used the
phyllotaxis raster grid in their art work.
Figure 2.15. Phyllotaxis raster grid
Paturi, the Austrian researcher and author of the remarkable book, Plants as
Ingenious Engineers of Nature [140], made an analysis of the use of phyllotaxis
raster grids in the artworks of great artists. For this purpose, he put together the
phyllotaxis raster grid with Titian’s picture Bacchus and Ariadne (Fig. 2.16).
After an analysis of Titian’s painting Paturi concluded:
“All basic lines of the perspective coincide with the raster grid. Even the
set of the minor parts and forms were placed by the artist in that field of
internal constraints, on which the picture is constructed. Pay attention to
the line along the small hill seen on the horizon near to the church campa
nile, to the branches of the large tree, to the outline of the cloud lying under
the constellation, to the hind paws and the belly line of the large wild cat,
Alexey Stakhov
THE MATHEMATICS OF HARMONY
110
Figure 2.16.
Titian’s Bacchus and Ariadne
along the direction of the axis of the over
turned vase, to the raised right hand of the
satyr in the garland of grapevines in the
right corner of the canvas and, at last, to
the raised leg of the horse.”
Paturi further concluded:
“At all times the artists consciously or un
consciously comprehended the laws of aesthet
ic perception by watching nature. Artists were
always enchanted by the simple and simulta
neously rational geometry of the growth of bi
ological forms.”
2.13. “Fibonacci Resonances” of the Genetic Code
2.13.1. The Initial Data about the Genetic Code
Among the biological concepts [141] that are well formalized and have a
level of general scientific significance, the genetic code takes special prece
dence. Discovery of the striking simplicity of the basic principles of the genet
ic code places it amongst the major modern discoveries of mankind. This sim
plicity consists of the fact that inheritable information is encoded in the texts
from threelettered words — triplets or codonums compounded on the basis
of the alphabet that consists of the four characters or nitrogen bases: A (Ade
nine), C (Cytosine), G (Guanine), T (Thiamine). The given system of the ge
netic information represents a unique and boundless set of diverse living or
ganisms and is called the Genetic Code.
2.13.2. DNA SUPRAcode (JeanClaude Perez’s Discovery)
In 1990 JeanClaude Perez, an employee of IBM, made a rather unexpected
discovery in the field of the genetic code. He discovered the mathematical law
that controls the selforganization of bases A, C, G and Т inside the DNA. He
found that the consecutive sets of the DNA nucleotides are organized in frames
of remote order called “RESONANCES.” Here, the resonance means a special
proportion that divides the DNA sequence according to Fibonacci numbers
(1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 …).
Chapter 2
Fibonacci and Lucas Numbers
111
The key idea of Perez’s discovery, called the DNA SUPRAcode, consists of
the following. Let us consider some fragment of the genetic code that consists of
the A, C, G and T bases. Suppose that the length of this fragment is equal to
some Fibonacci number, for example, 144. If a number of the Tbases in the
DNA fragment is equal to 55 (Fibonacci number), and a total number of the C,
A and G bases is equal to 89 (Fibonacci number), then this fragment of the
genetic code forms is a resonance — that is, a proportion between three adjacent
Fibonacci numbers (55:89:144). Here it is permissible to consider any combina
tions of the bases, that is, C against AGT, A against TCG, or G against TCA. The
discovery consists of the fact that the arbitrary DNAchain forms some set of
the resonances. As a rule, the fragments of the genetic code of the length equal
to the Fibonacci number Fn are divided into the subset of the Tbases, and the
subset of the remaining A, C, G bases; here the number of Tbases is equal to the
Fibonacci number Fn2 and the total number of the remaining A, C, G bases is
equal to the Fibonacci number Fn1, where Fn=Fn1+Fn2. If we make a systematic
study of all the Fibonacci fragments of the genetic code, we can obtain a set of
the resonances that is called the SUPRAcode of DNA.
2.13.3. A Verification of JeanClaude Perez’s Law
In the Petoukhov book [141] the sequences of triplets for the α and β
chains of insulin are given. For the βchain this sequence has the following form:
ATGTTGGTCAATCAGCACCTTTGTGGTTCTCACCTCGTT
GAAGCTTTGTACCTTGTTTGCGGTGAACGTGGTTTCTTC
TACACTCCTAAGACT
Note that all the Tbases in the indicated sequence are marked by bold type.
A verification of JeanClaude Perez’s Law using the βchain of the insulin
molecule as an example (see above) results in the following outcome. The total
number of the triplets in the βchain is 30, that is, the molecule contains 90 bases
(the nearest Fibonacci number is 89). If we count the number of Tbases in the
above βchain, we find that there are 34 (a Fibonacci number). Then the number
of the remaining A, C, G bases is equal to 9034=56 (the nearest Fibonacci num
ber being 55). Thus, there is the following proportion between the Tbases and
the rest, i.e. the A, C, G bases in the βchain are 90:56:34. This proportion is very
close to the Fibonacci resonance of 89:55:34. It follows from this analysis that
JeanClaude Perez’s Law for the insulin βchain is fulfilled with great accuracy. If
now we take the initial segment of the above βchain that consists of the first 18
triplets, that is, of the 54 bases (the nearest Fibonacci number is 55), and count
the number of Tbases in this fragment we find that it is 22 (the nearest Fibonacci
Alexey Stakhov
THE MATHEMATICS OF HARMONY
112
number being 21). This means that we have the following proportion in the first
fragment of the βchain, namely, 54:32:22 that is close to a Fibonacci resonance of
54:34:21.Thus, the Perez’s Law is also fulfilled for the first fragment. If we take the
segment that consists of the rest of the 12 triplets (36 bases) then the number of
the Tbases in this segment is 12 (the nearest Fibonacci number being 13). Thus,
for this case we have a proportion of 34:24:12 that is close to the Fibonacci reso
nance of 34:21:13. Therefore, both for the βchain of the insulin molecule as a
whole, and for its separate fragments, the Perez Law is fulfilled in practice with
sufficient accuracy. In addition, it is possible to see that practically in any segment
of the βchain there is a tendency towards the golden mean.
This surprising discovery of JeanClaude Perez has a number of interest
ing applications in the socalled plastic arts and market analysis. Below we
can see application of this law in, for example, music, poetry, cinema and mar
ket processes (Elliott Waves).
2.14. The Golden Section and Fibonacci Numbers in Music and Cinema
2.14.1. Pythagorean Theory of Musical Harmony
Music is a form of art reflecting reality and influencing a person by means of
organized sound sequences. Preserving some similarity to natural sounds, musical
sounds principally differ from the latter by constrained pitch and rhythmic organiza
tion (musical harmony). Since antiquity, the search for the “laws of musical harmony”
has been one of the important pursuits of science. Greek musicology does not corre
spond exactly with the character of modern musicology. Greek musicology was not
directed toward analyzing musical works. Rather, the Greeks saw musicology as a
study of the acoustic aspects of sound. An essential feature of Greek musicology was
the aspiration for a mathematical description of musical harmony.
Pythagoras is credited with the discovery of two fundamental harmonic
musical laws:
1. If the ratio of the oscillation frequencies of two tones is describable in
small integers, a harmonic sound may result;
2. To get a harmonic triad, it is necessary to add a third tone to the chord,
which consists of two consonant tones. The oscillation frequency of the
third tone should be in harmonic proportional connection to the two tones
of the chord.
Pythagoras’ work on the scientific explanation of musical harmony was the first
Chapter 2
Fibonacci and Lucas Numbers
113
wellgrounded scientific theory of musical harmony! Pythagoras attempted to apply
his musical theory to cosmology; according to his ideas, the planets of the Solar Sys
tem are arranged pursuant to the musical octave (“harmony of the spheres”).
There are many legends surrounding the acoustic experiments of Pythag
oras. The most popular of them describes how Pythagoras, passing by a black
smith, heard the sounds of hammer blows on an anvil and found in them an
octave, a fifth and a fourth. In his excitement, Pythagoras hastened to the
smithy and after a series of experiments with hammers, found that the differ
ence in sounds depended on the weight of the hammers.
The Pythagorean discovery of the deep connection between music and
mathematics caused harmonics to be included amongst the mathematical sci
ences influencing the future development of musical theory.
2.14.2. Chopin’s Etudes in the Lighting of the Golden Section
Any piece of music has a temporal duration and is divided into separate
parts by marks that attract our attention (“expression marks”) and simplify
our perception of the musical work. These marks are dynamic culmination
points within the musical work. The question is whether or not there is cer
tain regularity in the placing of “expression marks” in the musical work. An
attempt to answer this question was made by Russian musicologist Sabaneev.
He showed [142] that the separate time intervals of the musical works of
Chopin are connected, as a rule, by “culmination events” dividing the musical
work in the golden ratio. Sabaneev writes:
“All such events are referred by the author’s instinct to such points of the
musical work that divide the temporal durations into separate parts being in
the ratio of the “Golden Section.” Observation shows that quite often similar
“expression marks” are found at golden points. It is quite surprising, because
frequently the knowledge about these things is absent for many composers,
and these facts are a consequence of internal feelings of rhythm.”
Sabaneev’s analysis of a large number of musical works allowed him to con
clude that the organization of music is often done so that its cardinal parts, divided
by the “marks,” are based upon the golden mean series. Such organization of music
corresponds to the most economical perception and consequently creates an im
pression of the best “regularity” within the musical structure. In Sabaneev’s opin
ion, quantity and frequency of golden mean appearances in music depends upon a
“composer’s expertise.” The musical work of great composers is distinguished by
the highest percentage of golden mean appearances, that is, “the intuition of the
form and regularity, as would be expected, is highest for the first class geniuses.”
Alexey Stakhov
THE MATHEMATICS OF HARMONY
114
According to Sabaneev, the golden mean is repeatedly met in the musical
works of different composers. Every such golden point reflects the overall mu
sical event, the qualitative jump in the development of the musical theme. In
the 1,770 musical works of 42 composers that were studied by him, the golden
mean is met 3,275 times. The greatest frequency of musical works based upon
the golden mean is found in Beethoven (97%), Haydn (97%), Arensky (95
%), Chopin (92%), Mozart (91%), Schubert (91%), and Scriabin (90 %).
Chopin’s 27 etudes were studied by Sabaneev in particular detail. Golden
sections were found 154 times in them; the golden section being absent in
only three etudes. In some cases, the principles of symmetry and the golden
mean are combined simultaneously in the structure of a musical work; in these
cases, the musical work is divided into some symmetrical parts; however, the
golden section is met within every separate part. For example, Beethoven’s
music is often divided into two symmetrical parts; however, the golden sec
tion is always met inside each part.
2.14.3. Rosenov’s Research
The Russian art critic Rosenov paid particular attention to research into
the harmonic laws of music. He asserted that the stringent proportional rela
tions in music and poetry that are present with the golden mean should play
an outstanding role.
Rosenov chose for his analysis a number of typical musical works of out
standing composers: Bach, Beethoven, Chopin and Wagner. For example, by
studying Bach’s Chromatic fantasy and fugue, he used the quarter duration as
the unit of measure. There are 330 such units of measure in this musical work.
The 204th quarter from the beginning corresponds to the golden section here.
This golden point coincides precisely with the fermata (in musical pieces the
fermata sign augments a pause’s duration, usually in 1.52 times). Further
more, the golden point separates the first part of the musical work (the pre
lude) from the second part. The fugue, which follows the fantasy, also demon
strates an astonishing harmony of parts. Looking at the scheme of the fugue’s
harmonic analysis, “one experiences a sacred thrill [and] comes into contact
with the greatness of the composer, who embodies the innermost laws of cre
ative musical work with such accuracy.”
Rosenov analyzed the final of the sonata of Cismoll by Beethoven, the
FantasiaImpromptu by Chopin, the introduction to Tristan and Isolda by
Wagner and so on. The golden section occurs very often in all of these musical
works. He paid special attention to Chopin’s Fantasia, which was an impromptu
Chapter 2
Fibonacci and Lucas Numbers
115
creation and not subject to any editing. Since this was impromptu, there was
no conscious application of the golden section, which nevertheless is present
throughout this musical piece even in the smallest musical parts.
Thus, we should recognize the golden section as a harmonic criterion in
many musical works appearing frequently. Every part can be divided into two
parts by the golden section and each new part can be divided by the golden
section. Here we can see an analogy with Fibonacci resonances of the genetic
code where each segment of the genetic code is divided in the golden ratio!
2.14.4. Bella Bartok’s Music
The Hungarian Bella Bartok is one of the composers who consciously used the
golden section in his music. The deep harmonic analysis of Bartok’s music was giv
en by Jay Kappraff [47] who wrote: “Bartok based his music on the deepest layer of
folk music. He believed that all folk music of the world can ultimately be traced to
a few primeval sources.” According to the Hungarian musicologist Erno Lendai,
“[Bartok] discovered and drew into his art the laws governing the depths of the
human soul which have been untouched by civilization.” As Kappraff noted, “Bar
tok based the entire structure of his music on the golden mean and Fibonacci series
— from the largest elements of the whole piece, whether symphony or sonata, to the
movement, principal, and secondary themes and down to the smallest phrase.”
2.14.5. The Golden Mean and Fibonacci Numbers in Cinema
Sergey Eisenstein (18981948) is the great Soviet film director and art theo
rist. Having seen inexhaustible new opportunities for art in cinema, Eisenstein soon
created his first film, Strike. In 1925 Eisenstein created the film The Battleship
Potemkin that caused a triumph in the world. The American Film Academy recog
nized The Battleship Potemkin as the best film of 1926. At the Parisian Exhibition of
Arts this film received the highest award: “Super GrandPrix.” It became a classic of
new cinema art. Creating the films October, Old and New, etc., Eisenstein developed
a theory of “intellectual cinema” that gave the spectator an opportunity to get not
only the artistic, but also scientific knowledge and concepts.
Eisenstein was a pioneer in the use of the golden mean in cinema. It is most
interesting how that Eisenstein used the “Golden Section Principle” in his film The
Battleship Potemkin. First of all this film consists of 5 acts. The partitioning of the
film into two separate parts corresponds to the proportion 2:3, that is, the first Fi
bonacci approximation to the golden mean. This 2:3 watershed occurs between the
end of the second and the beginning of the third act of the fiveact film, the basic
caesura of the film occurs as a Zero Point of the action stop. However, perhaps, the
Alexey Stakhov
THE MATHEMATICS OF HARMONY
116
most curious fact in all of this is that the “Golden Section Law” in The Battleship
Potemkin is observed not only for the zero point of the movement, but also is true for
the apogee point: a Red Flag on the battleship mast. Finally, the moment of the
appearance of the red flag is the golden section of the film’s duration.
2.15. The Music of Poetry
In many respects, poetic works are close to music. A distinctive rhythm,
regular alternation of stressed and unstressed syllables, a certain dimension
ality, and their emotional saturation make poetry a native sister to music. Each
verse has its musical form, rhythm and melody. It is possible to expect some
musical features in poems, for example, regularity of musical harmony, and
consequently, the golden section can often be present.
2.15.1. Pushkin’s Poetry
Alexander Pushkin was a Romantic author consid
ered to be the greatest Russian poet and the founder of
modern Russian literature. Pushkin’s father came from a
distinguished family of the Russian nobility. His moth
er’s grandfather was Abram Petrovich Gannibal, an Ethi
opian, who as a child was abducted by the Turks during
their governing of Ethiopia, and he became a great mili
tary leader, engineer and nobleman in Russia under the
Alexander Pushkin
auspices of his adoptive father, Peter the Great.
Born in Moscow, Pushkin published his first poem at the age of fifteen. By
the time he finished the first grade of the prestigious Imperial Lyceum in Tsar
skoe Selo near St. Petersburg, his talent was widely recognized by the Russian
literary community. In 1820 he published his first epic poem, Ruslan and Lyud
mila. Because of his 1820 poem, Ode to Liberty, he was exiled by the Czar Alex
ander I to the south of Russia. First coming to Kishinev in 1820, he there be
came a Freemason. He was in Kishinev until 1823. After a summer trip to the
Caucasus and to the Crimea, he wrote two Romantic poems which brought him
wide popularity, The Captive of the Caucasus and The Fountain of Bakhchisaray.
When Alexander’s brother, Nicholas I, came to power in 1825, he invited Push
kin back to the capital and gave him a government post. However, Nicholas acted
as his personal censor to make sure that Pushkin did not publish anything that
Chapter 2
Fibonacci and Lucas Numbers
117
would hurt the government. The Czar ordered his spies to follow him, cutting out
whole stanzas from his manuscripts. In the autumn of 1830 Pushkin left the cap
ital to visit a small village left to him by his father. There Pushkin wrote some of
his best poems, including completing his most famous poem Eugene Onegin.
On January 19, 1831, when he was almost 30 years old, Pushkin married
the beautiful young Nathalie Goncharova. Although they had three children,
they were not a happy couple. Nathalie was very beautiful and a favorite at
the court, often encouraging the attention of other men. In 1837, influenced
by rumors that his wife had entered into a scandalous liaison, Pushkin chal
lenged her alleged lover, Georges d’Anthиs, to a duel. Pushkin was mortally
wounded in the duel and died on January 29, 1837.
Pushkin’s poetic works were analyzed by many researchers from the gold
en mean point of view. Let us begin with the poem’s size, the number of lines.
At first, it seemed that this parameter of the poem could change somewhat
arbitrarily. However, this proved not to be so. For example, the analysis of
Pushkin’s poems by Vasjutinsky [31] showed that the sizes of Pushkin’s po
ems are not distributed uniformly; Pushkin appearantly preferred poems of 5,
8, 13, 21 and 34 lines (Fibonacci numbers!).
It was noted by many researchers that his poetical verses and poems are sim
ilar to musical pieces; there are also culmination points in them dividing the po
ems at the golden section. By studying Pushkin’s poem Shoemaker, Vasjutinsky
noted that the poem consists of 13 lines. Two semantic parts can be singled out in
the verse: the first part consists of 8 lines and the second one (the parable moral)
consists of 5 lines. Note of course that the numbers 13, 8, 5 are Fibonacci numbers
and their ratios are thus an approximation to the golden mean proportion!
One of Pushkin’s last poems, Not Dearly I Appreciate the HighSounding Rights
…, consists of 21 lines in which two semantic parts are singled out: the first part is 13
lines and the second part 8 lines (21, 13 and 8 are of course Fibonacci numbers).
The analysis of the famous novel, Eugene Onegin, by Vasyutinsky [47] is very
intriguing. This novel consists of 8 chapters; each chapter is, on the average, about
50 verses. The eighth chapter is the most perfect and the most emotional. This
chapter consists of 51 verses. Together with the letter of Eugene to Tatjana, the
size of this chapter corresponds precisely to the Fibonacci number 55.
Vasyutinsky writes: “Eugeny’s declaration of love to Tatjana is the chap
ter’s culmination (the line ‘To turn pale and to die away… it is bliss!” This line
divides the 8th chapter into two parts, the first part consists of 477 lines and
the second part consists of 295 lines. Their ratio is equal to 1.617! We can see
the finest conformity to the golden mean! This fact is the great miracle of
harmony created by Pushkin’s genius!”
Alexey Stakhov
THE MATHEMATICS OF HARMONY
118
2.15.2. Lermontov’s Poetry
Rosenov analyzed many poetic works of Lermontov, Schiller and Tolstoy,
also finding in them the golden mean.
Michael Lermontov was born on the 15th of Octo
ber, 1814 into a family of nobility. The poet spent his
youth at Tarkhany. He enrolled in Moscow University,
but very soon had to leave the University. Later he en
rolled in and graduated from St. Petersburg School of
Cavalry Cadets. In 1837, he was exiled to the Cauca
sus because of his poem on Pushkin’s death, in which
he blamed the ruling circles of Russia and the Czar Nico
las I. In 1841, Lermontov was sent into exile in the Cau
Michael Lermontov
casus once again. As a result of intrigues between the
officers, he was provoked into a personal quarrel with an old schoolfellow.
This led him to a duel on July 15, 1841 in which the poet was killed.
Lermontov began writing his poems when he was very young. However, he
became famous because of his poem on Pushkin’s death. Lermontov’s poems,
including The Demon and Mtsyri, and his innumerable lyrical poems such as, A
Hero of Our Time and Masquerade, are masterpieces of Russian literature.
Lermontov’s famous poem, Borodino (see English translation by Eugene
M. Kayden at www.lermontov.net/content/view/39/2/), is divided into two
parts: the short introduction and the main part. The introduction consists
of only one stanza:
“But tell me, uncle, why our men
Let Moscow burn, yet fought again
To drive the French away?
I hear it was a dreadful fight,
A bitter war, by day and night;
That’s why we celebrate the might
Of Borodino today.”
The main part of the poem consists of 13 verses, and it is divided into
two parts: first consist of 8 verses, and second of 5 verses. In the first part, a
battle of Borodino is described with rising tension an expectation of action;
in the second part, the battle itself is described with gradual reduction of
tension to the end of the poem. Each verse consists of 7 lines, that is, in total
the main part consists of 91 lines. If we divide the main part in the golden
section (91/1.618 = 56.238), we find, that the division point corresponds to
the beginning of the 57th line, that begins from the short phrase: “O what a
Chapter 2
Fibonacci and Lucas Numbers
119
day!” This phrase represents the culmination point of the tense anticipa
tion, which completes the first part of the poem the expectation of the
action, and opens its second part a description of the battle:
O what a day! The Frenchmen came,
A solid mass, like clouds aflame,
Straight for our redoubt.
Their lancers rode with pennons bright;
Dragoons came on in all their might
Against our walls, and in the fight
They scattered in a rout.
2.15.3. Shota Rustaveli’s Poem
Shota Rustaveli was a Georgian poet of the 12th century.
He is one of the greatest representatives of medieval literature.
There is not much biographical data about Rustaveli: the ex
act dates of his birth and death, and historical data about the
main events of his life are unknown. Many poetic works writ
ten by him were lost. He is the author of the literary work The
Knight in the Panther’s Skin, a Georgian national epic poem.
Rustaveli was a Georgian noble and treasurer to the Queen of
Georgia Tamara. He also restored and painted frescoes in the Shota Rustaveli
Georgian monastery of the Holy Cross in Jerusalem. One of
the pillars of this monastery bears a portrait, which is believed to be that of the
poet. According to the legend, he fell hopelessly in love with Tsarina Tamara. It
was here in the cell of this monastery that Rustaveli terminated his life.
A grandiose poem, The Knight in the Panther’s Skin, brought to Rustaveli
world glory. Translated into many languages this poem is rightfully consid
ered one of the world’s greatest pieces of literature.
Many researchers of Shota Rustaveli’s poem note its harmony and melody.
In the opinion of Georgian scientist and academician Zereteli, these properties
of Rusataveli’s poem arise, thanks to his conscientious use of the golden mean in
both its poetic forms and the construction of its verses. Rustaveli’s poem con
sists of 1 637 stanzas; each consisting of four lines. Each line consists of 16 sylla
bles divided into two equal parts, the halflines, which therefore consist of 8
syllables. All halflines are divided into two segments of two kinds: the Akind
segment is the halfline with equal segments and with an even number of sylla
bles (4+4), the Bkind segment is the halfline with asymmetrical division into
two unequal parts (5+3 or 3+5). Thus, we can see the following 3:5:8 ratio in
Alexey Stakhov
THE MATHEMATICS OF HARMONY
120
the halfline of the Bkind; that is, Fibonacci numbers, whose ratios are, of course,
close to the golden mean. Zereteli proved that 863 of the 1 637 stanzas in Rustave
li’s poem are constructed according to the golden mean principle!
2.16. The Problem of Choice: Will Buridan’s Donkey Die?
The Russian academician Zeldovich wrote in one of his works: “Do not
treat with contempt “simple” reasoning. Those researchers, who obtain deep
results out of the simple, but firmly established facts, deserve the highest
praise.” The works of American psychologist Vladimir Lefevre have an utter
“simplicity” of initial reasoning giving them aesthetic charm.
2.16.1. A Parable about Buridan’s Donkey
Often times we have to choose the best amongst various possibilities. This
leads to a rather unexpected consequence — the pleasure of choice suddenly
turns into a complex problem, and this problem can be extremely serious. This
problem of choice is a perennial problem. The Greek philosopher Aristotle wrote
of the difficulty for a person experiencing both hunger and thirst who is equally
distant from food and drink. He remains unmoved at the same place because
he cannot make a choice. This problem of a person not being able to make a
choice between food and drink, was formulated more precisely by the French
philosopher Jean Buridan, who demonstrated an apparent absence of the free
dom of will. He gave the example of the donkey who, being situated between
two equally sized haystacks each equally distant, should by all means die when
the donkey cannot choose which haystack to go and eat. Since then a popular
expression, Buridan’s donkey, has arisen. A person, irresolute in a choice or
vacillating between two equivalent situations, is called “Buridan’s donkey.”
2.16.2. A Psychological Experiment
We are quite confident that repeatedly throwing a coin inevitably results in
50% heads and 50% tails. American psychologist Vladimir Lefevr began to re
flect upon whether one can apply this result to psychology. For this reason, he
carried out the following experiment. He asked a person to divide a pile of string
beans into two piles; the good string beans are in one of them and the bad string
beans in the other. String beans in the initial pile are all very similar to one
Chapter 2
Fibonacci and Lucas Numbers
121
another, and it is difficult to give any objective criterion for their division into
the “good” and “bad” string beans. It is clear that this problem is similar to throw
ing a coin and we can expect the result: 50% by 50%. However, the actual result
overturned the expectations: the number of the good string beans steadily built
up to 62% (0.62) from the initial number of string beans. What a surpise! As we
know, 62% is the golden mean of 100%! The result of the psychological experi
ment proved very surprising: a person divides the string beans into two piles in
the golden ratio! Vladimir Lefevre offered an original explanatory theory for
this experiment. However, to understand Lefevre’s theory, it is necessary to know
some basic concepts of mathematical logic and psychology.
2.16.3. What is an Implication?
The name of the English mathematician George Boole (18161864) is wide
ly known in modern science. In 1854 he published An Investigation of the Laws
of Thought, on Which are Founded the Mathematical Theories of Logic and Prob
abilities. His work produced the beginning of a new algebra, the algebra of logic,
or Boolean Algebra. Boole showed for the first time, that there is an analogy
between algebraic and logical operations, if we assume that logical variables
take only two values — true or false (symbolized as 1 or 0). He invented a sys
tem of designations and rules, which allows one to code any statements and
then to operate on them as traditional numbers. Boolean algebra has three basic
operations — AND, OR, NOT, which allow one to perform logical operations of
conjunction, disjunction and negation on statements. In his Laws of Thought
(1854) Boole finally formulated the basis of mathematical logic. However, as
often happens with mathematicians, Boole wrote that he did not see the practi
cal applications of his algebra. Fortunately, regarding practical applications of
his algebra the great mathematician was mistaken. Modern science, in particu
lar computer science, is impossible without Boolean logic, which is widely used
in the analysis and synthesis of digital automatons, as well as in the solutions of
logic problems on computers.
Propositions can be 1 (true) or 0 (false) and Boolean Functions connecting
them are the basis of Boolean algebra. A proposition is any statement that can
be evaluated from the point of view of its truth or falsity. As an example, con
sider the two propositions:
A = “George Boole is the creator of Boolean algebra”
B = “2×2=5”
The first proposition is true, that is, A=1, the second proposition is false,
that is, B=0.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
122
Propositions can be Simple or Complex. We gave examples above of simple
propositions. Their basic property is the fact that they can be either true or
false. The complex proposition consists of several simple propositions, thus,
the complex proposition is also either true or false (1 or 0). However, the truth
value of the complex proposition depends on the truth values of the simple
propositions that make up the complex proposition. Thus, the complex prop
osition Y is a logic function (Boolean function) of the simple propositions
contained within it (for example, A and B), that is, Y=ƒ(A,B).
The following logic functions — Negation, Conjunction (the logic function
AND), Disjunction (the logic function OR), Addition by the Module 2, etc. are
the most widespread elementary Boolean functions. Boolean functions are
employed in computers. Boolean functions are also used in other fields of
knowledge including implication in psychology. Implication is a logical oper
ation that builds up the complex proposition corresponding to the logical con
nective “if … then.” Implication consists of two simple propositions, the Ante
cedent which follows “if” and the Consequent which follows “then.” If A and B
are the antecedent and the consequent, respectively, then the implication is
the Boolean function Y=A→B that is false (Y=0) only in the case where the
antecedent is true (A=1) and the consequent is simultaneously false (B=0);
for the rest of the cases the implication is true (Y=1).
The implication is represented by Table 2.14.
Table 2.14. A truth table for implication
What would be a physical applica
tion
of the implication? How can the
A
0
0
1
1
B
0
1
0
1
implication connect two simple propo
Y
1
1
0
1
sitions? We will show this in the exam
ple of the statements: A = “the given quadrangle is a square” and B = “around a given
quadrangle we can describe a circle.” Let us examine the complex proposition A→B
= “if the given quadrangle is a square, then around the given quadrangle we can
describe a circle.” There are three variants, when the proposition A→B is true (1):
1. A is true (1) and B is true (1), that is, a given quadrangle is a square
and it is possible to describe a circle around it.
2. A is false (0) and B is true (1), that is, a given quadrangle is not a
square, however, it is possible to describe a circle around it (it is clear that
the proposition B is not true for all quadrangles).
3. A is false (0) and B is false (0), that is, a given quadrangle is not a
square and it is not possible to describe a circle around it.
Note that only one variant for the implication is false (0), when A is true
and B is false, that is, when a given quadrangle is a square, and at the same
time, it is not possible to describe a circle around it.
Chapter 2
Fibonacci and Lucas Numbers
123
In usual speech a connective “if ... then” describes a relationship of cause
and effect between propositions. However, in terms of logical operations the
meanings of the propositions are not considered. We only consider the truth
or falsity of the propositions. Therefore, we should not be confused by the
seeming “nonsense” of the implications formed by the propositions that can
not be connected by their contents. For example: “If the President Bush is a
democrat, then giraffes are in Africa”, “if a watermelon is a berry, then gasoline
is sold at the gas station.”
2.16.4. What is a Reflection?
In the most general sense reflection maybe though of as the human soul
pondering itself. There is variety of definitions for this concept. More often, a
reflection is determined as the analysis of one’s own ideas and experiences; a
reflection is full of doubts and uncertainties. For the first time the concept of
Reflection in a contemporary sense was used by the British philosopher John
Locke (16321704). From his point of view, a reflection is a special operation
by a person upon his own consciousness. As a result, this operation generates
ideas about one’s own consciousness. Further down, in relation to a person
words “his” and “himself” stands for both genders.
A person is reflecting, when one says: “I think that I think,” and so forth.
According to Lefevre’s opinion, a description of reflection signifies that a person
a0 can be represented symbolically by two elements: a person and his “map of
consciousness,” in which the information about how the person thinks about him
self is placed. This means that the “map of consciousness” contains some image Z
of a person regarding himself (Fig. 2.17).
There therefore appears to be a reflective struc
ture, in which the person possesses a “map of con
sciousness,” where the person’s image about himself
is represented. Symbolically such a structure can be a0
a
Figure 2.17. A person with
expressed by the formula A = a01 , where a0 is the
person himself (how the “absolute observer” can see his “map of consciousness” Z
a person from outside), and a1 is the image of a person about himself. On the other
hand, it is possible toaconsider
the reflection of the
a 2
second order A = a01 , where a1 is the image of a
person thinking about himself, and a2 is the image
of a person about his own thoughts (Fig. 2.18).
a
1
a0
Now, let us introduce one more intriguing
Figure 2.18. A reflective
word, Intention (from the Latin word intentio —
structure of the second order
Z
Z
Alexey Stakhov
THE MATHEMATICS OF HARMONY
124
aspiration). This word means an aspiration, a purpose, a direction or an orienta
tion of consciousness, wish, and feelings towards any subject. In the expression
a0a1 the symbol a0 means the intention of the person and the symbol a1 means
the intention of his image. We can estimate the intentions and the acts of the
person as positive (1) or negative (0) as done in Boolean algebra. A negative
estimation of oneself means the presence of guilt. A negative estimation of the
partner means the feeling of the partner’s guilt and selfcondemnation.
Let us consider the expression:
a
a 2
(2.77)
A = a01 .
a2
Here, the power A1 = a1 means the image of the person in his representa
tion. The appraisal A1=0 means that a person estimates himself negatively,
vice versa, A1=1 means that a person estimates himself positively. The power
a2=1 is a representation of what a person thinks about how he estimates him
self. The appraisal a2=1 means that a person has a positive selfappraisal. The
appraisal a2=0 means that a person has a negative selfappraisal.
Lefevre started to study this kind of selfreflection (2.77), that is, the reflec
tion of the second order, where a1 is a representation (a thought) of a person about
himself, a2 is a representation (a thought) of a person about his own thought.
Lefevre’s important idea consists of the fact that the appraisal of a one’s
self and negative or positive determination of this appraisal is fulfilled by a
person without any efforts from the person’s consciousness, that is, each per
son seems to have a “reflective computer” that automatically makes these self
appraisals according to the formula (2.77).
Now consider the values taken by the function (2.77) with dependence on
the variables a0, a1, a2. With this purpose in mind, we first take into account
the expression. We can write the numerical values of this expression with de
pendence on the values of a1 and a2 as follows:
11=1; 01=0; 10=1; 00=1.
(2.78)
These expressions do not raise any objections from the mathematical point of
view. In mathematics, the expression 00=1 is accepted as “correct” by definition.
Now compare the expressions (2.78) with the truth table for implication
a
(Table 2.13). It follows from the comparison that the expression A1 = a1 2 sets
the implication A1=a2→ a1. If we consider the values of the function (2.77), we
can conclude that this function creates the implication of a more complicated
kind, namely:
(2.79)
A=(a2→ a1)→ a0.
Note that the expression (2.79) is Lefevre’s key idea! Let us consider a
truth table for this logical function (2.79).
Chapter 2
Fibonacci and Lucas Numbers
125
Table 2.15. A truth table for Lefevre’s function
ao
1
1
1
1
0
0
0
0
a1
1
0
1
0
1
0
1
0
a2
1
1
0
0
1
1
0
0
А
1
1
1
1
0
1
0
0
Analysis of Table 2.15 results in very interesting conclusion. The total num
ber of possible values for Lefevre’s function (2.79) is equal to 8. Five of them
take the value 1 (a positive estimation) and the remaining 3 values take the
value 0 (a negative estimation). Note that 8, 5 and 3 are adjacent Fibonacci
numbers! This means that according to Table 2.15, our “reflective computer”
automatically (that is, without participation of our consciousness) gives a Fi
bonacci ratio between positive (5) and negative (3) estimations. This 5:3 ratio
is a Fibonacci approximation to the golden mean of 1.618! That is, Lefevre’s
theory explains an occurrence of the golden mean in the above psychological
experiment with the string beans, that is, a person automatically, without the
participation of his consciousness, tends to make a choice according to the golden
mean principle! This paradoxical conclusion can have very unexpected applica
tions to the socalled “behavioral” sciences. The theory of Elliott’s wave that
has actively developed in American science is one such theory.
2.17. Elliott Waves
However, if Lefevre’s theory is true, that is, all processes of the behavior of a
person and his decisions are based on (or related to) the “Golden Mean Princi
ple,” then perhaps Fibonacci numbers and the golden mean can be discovered in
such “behavioral systems,” such as in an economy that is a function of human
behavior through the decisions made by many people. Here we may be in for a big
surprise! In the first half of the 20th century the American bookkeeper and econ
omist Ralph Elliott developed an original theory about stock market price fluctu
ations. This research resulted in the modern science of Elliott Waves.
2.17.1. Ralph Nelson Elliott
Let us start from who was Elliott and how did he come by his discovery?
Ralph Nelson Elliott (18711948) was one of the brightest representatives of
the “American Renaissance.” Being an accountant by education, he special
Alexey Stakhov
THE MATHEMATICS OF HARMONY
126
ized in reorganizing and revitalizing large companies such as exportimport
houses and railroads in the U.S. and Central America. In 1924 the U.S. State
Department appointed Elliott to serve as Chief Accountant for Nicaragua
where the U.S. had large economic ties.
Later, Elliott wrote a foreign policy proposal based upon his longterm
experience in Central and South America and submitted it to the State De
partment. The essential ideas of his proposal were reflected in the program
“Good Neighbor” as elaborated by Roosevelt’s administration, and used later
in the more recent programs of the World Bank.
In the 1930s Elliott turned his attention toward studying the stock mar
ket. In 1938 at the age of 67, he published his first monograph about market
models, The Wave Principle. In the next year, he published a series of articles in
Financial World magazine that detailed his discovery. In 1946 he completed a
large book, Nature’s Law, in which he expanded his 1938 essay on the connec
tion between the Wave Principle, the golden mean and the stock market.
2.17.2. Rhythm in Nature
The proposition that the Universe is subordinated to some general Laws
returns us to Pythagoras and Plato, and the origin of this great universal recog
nition in modern science. It is obvious that without the Laws of Nature we would
have chaos. A strict ordering and constancy of Nature follows. This is confirmed
by a surprising periodicity and repetition in all of Nature’s processes.
A person is a natural object similar to the Sun or the Moon. One’s activity
can be expressed by the language of numbers and is subject to scientific analy
sis. Human activity, for example the heart’s activity, could be considered from
the point of view of rhythmic processes. A person’s heart beats uniformly (about
60 beats per minute). The heart acts as a cylinder piston by drawing in and then
pushing out the blood. Blood pressure changes during this cardiac cycle. It
reaches its greatest value in the left ventricle at the moment of compression
(systole). In arteries (during systole) the blood pressure is reaching its maxi
mum value, equal to 115125 mm Hg. At the moment of the cardiac reduction
(diastole), the pressure is decreasing until there is 7080 mm Hg. The ratio of
the maximum (systolic) pressure to the minimum (diastolic) pressure is equal,
on the average, to 1.6, which is a close approximation to the golden mean.
Is this a mere random coincidence, or does it reflect some objective regularity
of the organized cardiac activity? The heart beats continuously from a person’s
birth up to his death. And its activity should be optimal and be subordinate to the
selforganizational laws of biological systems. Because the golden mean is one of
Chapter 2
Fibonacci and Lucas Numbers
127
the criteria of selforganizing systems, one might naturally expect the cardiac cy
cle to be subordinate to the golden mean law. This hypothesis underlies the mam
mal cardiac research activity of Russian biologist Tscvetkov [39].
Judgments about heart activity are done through an electrocardiogram,
displaying a curve that reflects cardiac performance (Fig. 2.19).
In a cardiogram, a comparison is
made of two time intervals of different
duration corresponding to the heart’s
systolic (t1) and diastolic (t2) activity.
Tscvetkov found that there exists the
optimal (golden) frequency for people
and other mammals; here, the durations
of systole, diastole and the full cardiac
cycle (T) are in golden mean proportion,
that is, T: t2= t2: t1. For example, for a
person this golden frequency is equal to
F ORS
T
F ORS
T
63 heart beats per minute, and 94 beats
Figure 2.19. Human electrocardiogram
per minute for dogs.
Tscvetkov found that if we take the middle blood pressure in the aorta as
the measurement unit, then the systolic blood pressure is 0.382, and the dias
tolic pressure is 0.618, that is, their ratio corresponds to the golden ratio
(0.618:0.382=1.618). It means that cardiac performance in the timing cycles
and blood pressure variations are optimized according to the same principle,
the law of the golden mean.
If we move on from the investigation of the person as a biological unit to
the consideration of our social and economic activity, we find that human ac
tivity follows certain laws, which force the social and economic processes to
repeat in the form of a certain set of waves or impulses. The best way this idea
can be shown is through the example of stock exchange processes.
2.17.3. Elliott’s Wave Principle
Let us now return to a consideration of the main ideas concerning the
stock market that underlie the Elliott wave theory:
1. Natural Laws embrace the most important element of all, namely, tim
ing. They are not themselves some simple system or method for playing
the market. However, as expressed in market phenomena they appear to
mark the expression of all human activities. The application of these laws
to forecasting may have a revolutionary character.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
128
2. This law can only be discerned when the market is viewed and ana
lyzed as a product of human decisions. Simply put, the stock market is a
human creation and therefore reflects human idiosyncrasy.
3. All human activities have three distinctive features — pattern, time
and ratio — each of which appears to be based upon the Fibonacci series.
According to the Elliott Wave Theory, stock prices can be simulated by using
a specific wave that is connected with Fibonacci numbers. Elliott’s main wave is
shown in Fig. 2.20. More specifically, Elliott believed that the main wave is simu
lated by distinct upward and downward
a
5
movements. For example, the ascending
b
4
part of the wave consists of 5 movements,
c
three movements up and 2 movements
3
down (3 and 2 are of course adjacent Fi
2
bonacci numbers); the descending part of
1
the wave consists of 3 movements, 2 move
ments down and 1 movement up (again ad
Figure 2.20. Elliott Waves
jacent Fibonacci numbers).
The movements 1, 3 and 5 represent the Impulses in a Major Bull Move
ment. The movements 2 and 4 represent the Corrective Movements in a major
bull movement. The movements A, B and C define a Minor Bear Wave; here,
the movements A and C represent the descending movements of the minor
bear wave, while B represents one ascending movement of the minor bear wave.
The major waves determine the major trends of the market, and the minor
waves determine the minor trends. Elliott carefully examined the market’s
waves, and found that the golden mean and Fibonacci numbers play an im
portant role in the pressures and trends of the stock market.
Elliott proposed that the main wave exists at many levels; this means that
new subwaves could appear within the primary wave. To clarify, this means
that the chart above (Fig. 2.20) represents only the primary wave pattern. How
ever, the same kind of wave occurs, for example, between the points 2 and 4.
The diagram below (Fig. 2.21) shows how primary waves could be broken down
into smaller waves. We can see in Fig.
5 2
4
2.21 that the smaller waves are subor
a Bc
1
3
35
dinate to one and the same principle,
5 b
b
A
3
the golden mean principle.
C
a
c
1 2
4
4
Trading which is based upon the
2
Elliott Wave patterns is quite simple.
2
A trader identifies the main wave as
a supercycle and then acts accord
Figure 2.21. The smaller Elliott Waves
Chapter 2
Fibonacci and Lucas Numbers
129
ing to the Elliott Wave model, that is, increasing or decreasing the sales be
cause market behavior is subordinated to the golden mean principle. This goes
on in progressively shorter cycles until the cycle is complete.
Of course, real market processes are more complex than the Elliott model.
Even if we do not agree with some of Elliott’s findings, the idea is of great
methodological interest. Many scientists, especially in the U.S., are convinced
Elliott supporters. The American scientist Robert Prechter is the most fa
mous follower of Elliott’s ideas. In 1999, he published a book [143], which is
dedicated to the development of the Elliott Wave Principle, and organized
Elliott Wave International to promote Elliott’s ideas.
Prechter made the following very ambitious statement: “(R.N. Elliott’s)
Wave Principle is to sociology what Newton’s laws were to physics” [143].
Time will tell whether Prechter’s comparison of the Elliot Wave Princi
ple to Newton’s Laws has merit. However, one thing is doubtless: thanks to
the activities of Elliott and his followers, a theory of modern sociology and
market economics was added with deep scientific significance. According to
this concept, Fibonacci numbers and the golden mean are not only behind the
growth of the pinecone or seashell, but also determine the laws of human
behavior, and through them the laws of the stock market!
2.18. The Outstanding Fibonacci Mathematicians of the 20th Century
The 20th century is characterized by an increasing interest in Fibonacci
numbers and the golden section. Two scientific works, the book Aesthetics of
Proportions in Nature and Art [11] by the French scientist Matila Ghyka and
the book Proportionality in Architecture [10] by Russian architect Professor
G.D. Grimm were very important achievements in the “golden” area in the
first half of the 20th century.
In addition, considerable results in this area were obtained in the 20th Cen
tury by Danish mathematician Willem Abraham Wythoff and Belgian ama
teur mathematician Edouard Zeckendorf.
2.18.1. Willem Abraham Wythoff
Many specialists in combinatorial analysis and number theory know about
the Wythoff game. However, few know anything about the man for whom this
game was named. W. A. Wythoff was born in Amsterdam in 1865, the son of a
Alexey Stakhov
THE MATHEMATICS OF HARMONY
130
sugar refinery operator, and obtained his Ph.D. in mathematics in 1898 from the
University of Amsterdam. Wythoff’s game was described in his article A Modifi
cation of the Game of Nim [Nieuw Archief voor wiskunde, 2 (190507), 199202].
Dr. Wythoff described his famous game as follows:
“The game is played by two persons. Two piles of counters are placed on the
table, the number of each pile being arbitrary. The players play alternately and
either take from one of the piles an arbitrary number of counters or from both piles
an equal number. The player who takes up the last counter, or counters, wins.”
The solutions to Wythoff’s game that are based on Fibonacci numbers can
be found in numerous articles in The Fibonacci Quarterly, and also in the fa
mous article The Golden Section, Phyllotaxis, and Wythoff’s Game by Coxeter
(Scripta Mathematica, 19 (1953) 135143).
2.18.2. Edouard Zeckendorf
Many number theorists know about Zeckendorf sums,
however, here again, few know anything about the man
for whom these sums are named. Edouard Zeckendorf was
born in Liиge, Denmark. In 1925, he qualified as a medi
cal doctor at the University of Liиge and then became a
Belgian army officer. He also obtained a license for dental
surgery some time prior to 1930.
Mathematics nevertheless was Zeckendorf’s main pas
Edouard Zecken
dorf (19011983)
sion. In 1939, he published the article devoted to Zecken
dorf sums. According to his theorem, each positive integer has a unique repre
sentation as a sum of Fibonacci numbers, though two adjacent Fibonacci num
bers are actually never employed. Research on Zeckendorf sums became the
subject of many articles published in The Fibonacci Quarterly.
2.18.3. Verner Emil Hoggatt
The mathematical organization of the Fibonacci Associ
ation was founded in 1963 by a group of American mathe
maticians making it one of the most outstanding events in
the history of Fibonacci number theory. Beginning of 1963,
the Fibonacci Association began to issue The Fibonacci Quar
terly. The American mathematicians Verner Emil Hoggatt
(19211981) and Alfred Brousseau (19071988) were Verner Emil Hoggatt
founders of the Fibonacci Association. Verner Hoggatt, along
(19211981)
Chapter 2
Fibonacci and Lucas Numbers
131
with Brother Alfred Brousseau, published the first volume of The Fibonacci Quar
terly in 1963, thereby founding the Fibonacci Association. The April 4, 1969 issue
of TIME magazine reported the phenomenal growth of the Fibonacci Association.
In the same year, Houghton Mifflin published Verner Hoggatt’s book, Fibonacci
and Lucas Numbers [16], perhaps the world’s best introduction to Fibonacci num
ber theory. Howard Eves wrote that “during his long and outstanding tenure at
San Jose State University, Vern directed an enormous number of master’s theses,
and put out an amazing number of attractive papers. He became the authority on
Fibonacci and related numbers.”
2.18.4. Alfred Brousseau (19071988)
Alfred Brousseau was another outstanding person involved in the orga
nization of the Fibonacci Association. Brother Alfred belonged to the reli
gious order Fratres Scholarum Christianarum that translates as Brothers of the
Christian Schools, or simply, The Christian Brothers.
Brother Alfred was an avid photographer. He made a col
lection of some 20,000 slides of California wildflowers. Im
ages of more than half of these and other collections by Broth
er Alfred are preserved in the form of widely used websites at
the University of California, Berkeley.
In 1969, Time magazine featured two founders of the Fibonacci
Association in an article titled The Fibonacci Numbers. Brother
Alfred was pictured in the Time article holding a pineapple, one
of the best known representatives of phyllotaxis. The article re Alfred Brousseau
ferred to the many natural applications of Fibonacci numbers: for (19071988)
example, male bees reproduce “fibonaccically,” and Fibonacci numbers occur in the
formations visible in many sunflowers, pine cones, and leafpositions on branches of
trees. Brother Alfred recommended that people who learn about Fibonacci num
bers should focus their attention on the aesthetic pleasure involved in it.
2.18.5. Steven Vaida
In 1989, twenty years after Verner Hoggat’s book, the publishing house of
Ellis Horwood Limited published the book Fibonacci & Lucas Numbers and the
Golden Section [28] by Prof. S. Vaida. The book attracted wide attention from
Fibonacci mathematicians, as it is considered to be one of the best mathemat
ical books on the subject. Who is the author of this book? From his brief sci
entific biography, we learn that the mathematician Steven Vaida was Profes
sor of Mathematics of University Sussex (England) at the time of the book’s
Alexey Stakhov
THE MATHEMATICS OF HARMONY
132
publication. He obtained a Doctorate in Philosophy at Vienna University. In
1965, he began to work as Professor of operational researches at Birmingham
University. It is also noteworthy that he was an honorary member of many
mathematical organizations, in particular, the Mathematics and Statistics In
stitute of the Society of Operational Researches (England).
2.18.6. Herta Taussig Freitag
Among the modern Fibonacci mathematicians, it is necessary to name Pro
fessor Herta Taussig Freitag, another member of the Fibonacci Association.
She was born on December 6th, 1908 in Vienna, Austria and died January
25th, 2000 in Roanoke, Virginia.
Herta Freitag graduated with a Ph.D. from Columbia Uni
versity in 1953. Her colleagues at the Fibonacci Association
named Herta Freitag the Queen of the Fibonacci Association.
Over the years, she attended and presented a paper at every In
ternational Conference of Fibonacci numbers starting with the
first Conference in 1984. It is curious to note that Professor Fre
itag delivered the lecture Elements of Zeckendorf Arithmetic (co
Herta Taussig
author G.M. Phillips) at the 7th International Conference on
Freitag
(19082000)
Fibonacci Numbers and Their Applications. The appearance of
this lecture title is rather distinctive. It testifies to the fact that Fibonacci mathe
maticians came close to the creation of the new computer arithmetic based on
Fibonacci numbers and Zeckendorf sums. This became a central subject of the
Soviet scientific and engineering developments in the 1970s and 1980s.
2.19. Slavic “Golden” Group
The outstanding Hungarian geometer Janos Bolyai, one of the creators of
NonEuclidean geometry, once wrote the following remarkable words:
“For ideas, as well as for plants, the time comes, when they mature in their
various locales, just as in the spring the violets blossom wherever the Sun shines.”
The history of science should tell us why the 80s and 90s became the his
torical period in which interest in Fibonacci numbers and the Golden Section
began to peak. Particularly in this period, scientists of different disciplines
put forward hypotheses concerning the applications of the golden mean and
made discoveries that have fundamental significance for the development of
science and its many branches.
Chapter 2
Fibonacci and Lucas Numbers
133
The small brochure Fibonacci Numbers [13] published in 1961 by the Rus
sian mathematician Nikolay Vorobyov resulted in an expanding mathemati
cal interest in Fibonacci numbers. The brochure was reissued repeatedly and
translated into many languages.
First of all, it is necessary to mention the primary direction that arose in the 1970s
in Soviet science. It is well known that Fibonacci’s problem of “rabbit reproduction”
became a source for research into Fibonacci numbers. However, the “rabbit reproduc
tion problem” is not the only problem suggested by Fibonacci. He also proposed the
wellknown problem of Choosing the Best System of Standard Weights. This problem is
named the Weighing Problem or BashetMendeleev Problem in Russian historicmath
ematical literature. In 1977, the author of this book published the book Introduction
into Algorithmic Measurement Theory [20] and in 1979 the brochure Algorithmic Mea
surement Theory [21]. These works generalize the BashetMendeleev problem and
provide a new measurement theory based on Fibonacci numbers, namely Algorithmic
Measurement Theory. Published at the end of the 70s they helped determine the ap
plied nature of research in the Fibonacci field (Fibonacci codes, Fibonacci arithmetic,
Fibonacci computers) that began to develop in Soviet science. The practical direction
of Fibonacci research by the Soviet scientific school is distinctively different from the
direction of the American Fibonacci Association.
In 1984, two books were published dedicated to the golden section. The
Byelorussian philosopher Eduard Soroko in his book Structural Harmony of
Systems [25] made a courageous attempt to revive for modern science the
Pythagorean idea of the numeric harmony of the Universe, proposing the so
called Law of Structural Harmony of Systems that is expressed with the help of
the generalized golden sections, namely, the golden рsections [20]. In the
same year Stakhov’s book Codes of the Golden Proportion [24] was published.
This book developed number systems with irrational bases or codes of the
golden proportion that are a generalization of Bergman’s number system [86].
One year later in 1985, the book Aesthetic Foundations of Ancient Egyp
tian Art [145] by Russian art critic Natalia Pomerantseva was published; the
book demonstrates quite convincingly the role of the golden section in An
cient Egyptian culture.
The year 1986 was marked by the publication of two books on the Golden
Section. In Poland the EnergyGeometric Code of Nature [26] by Polish scien
tist and journalist Jan Grzedzielski was published. This book, probably for the
first time, uncovered the physical sense of the golden mean as the main code of
the Universe, as the proportion of thermodynamic equilibrium in selforganiz
ing systems. In the same year a teacher at the Kiev State Art Institute, Kovalev,
published a manual for artists called The Golden Section in Painting [29].
Alexey Stakhov
THE MATHEMATICS OF HARMONY
134
The beginning of the 1990s was characterized by the publication of two
more books on the Golden Section. The first was The Golden Section: Three
Approaches to the Nature of Harmony (1990) [146]. The book was authored
by three representatives of Russian art, the architect Shevelev, the composer
Marutaev and the architect Shmelev. The abstract states: “This book is dedi
cated to the theoretical substantiation of the phenomenon of the golden sec
tion, one of the universal laws of Nature.”
The popular book The Golden Proportion [31] by Nikolay Vasyutinsky, a
Ukrainian researcher, chemist, geologist, metallurgist and mechanic, was of
considerable importance of 1990. Written with great proficiency, it states: “the
manifestation of the Golden Proportion laws in architecture, music, poetry,
chemistry, biology, botany, geology, astronomy, and engineering sciences are
herein described.”
At the beginning of the 1990s, it became clear that Slavic science (Ukraine,
Russia, Belarus, and Poland) had developed an immensely powerful group of re
searchers formed of representatives of various sciences and arts, and authors of
very original books on the golden section. The idea arose to unite these research
ers together and create a Slavic golden scientific society. The First International
Seminar of The Golden Proportion and Problems of System Harmony was held in
Kiev in 1992 under the scientific supervision of Professor Stakhov, the author of
this book. The Belarussian philosopher Eduard Soroko (Minsk), the Ukrainian
architect and art critic Оleg Bodnar (Lvov), the Ukrainian mathematician and
economist Ivan Tkachenko, the Russian mechanical engineer Victor Korobko
(Stavropol), the Ukrainian physician and anatomist Pavel Shaparenko (Vinnit
sa), the Ukrainian chemist Nikolay Vasjutinsky (Zaporozhye), and the Polish
journalist Jan Grzedzielski (Warsaw) participated as members of the group’s or
ganizing committee. This group of highly respected scientists became the skele
ton of the Slavic scientists’ association called the Slavic Golden Group.
The Second International Seminar of The Golden Proportion and Problems
of System Harmony was held in 1993 in Kiev. Then on the initiative of Profes
sor Korobko, the Seminar continued its work in 1994, 1995 and 1996 in
Stavropol, Russia.
The Seminars stimulated research in the golden section in a variety of
fields. During the 1990s the Slavic Golden Group published a number of very
interesting books on the subject.
In 1994, the book The Golden Section and NonEuclidean Geometry in Na
ture and Art [37] was published by Oleg Bodnar. A new geometrical theory of
phyllotaxis (the Law of Spiral Biosymmetry Transformation) discovered by
Bodnar became the primary outcome of the book.
Chapter 2
Fibonacci and Lucas Numbers
135
The end of the 20th century was very successful for the Slavic Golden Group.
In 1997 Heart, Golden Section and Symmetry [39] by Russian biologist Tsvetk
ov was published. The book is a summary of the author’s longterm research in
the area of Golden Section applications to cardiac activity in mammals. In this
book the set of Golden Sections in the different structures of the cardiac cycle is
found. Tsvetkov also demonstrated the role of the Golden Section and Fibonac
ci numbers in optimization of the heart activity (minimization of energy con
sumption, in the blood, muscle and vascular matter) of mammals.
In 1998, the book The Golden Proportion and Problems of System Harmony
[43] was published by Professor Korobko (Russia), an active member of the
Slavic Golden Group. The book contains considerable information concerning
the application of the golden section throughout Nature, Science and Art. The
great strength of the book is that it can serve as a manual for university and
college teachers of postgraduate students in both the engineering sciences and
the liberal arts. In fact, the Russian Association for Building Universities rec
ommended it as a manual for all students in colleges and universities.
In 1999, Alexey Stakhov with assistance of Vinancio Massingue and Anna
Sluchenkova published Introduction into Fibonacci Coding and Cryptography
[44]. This book presented a new coding theory based on Fibonacci matrices.
In 2000, the famous Russian architect Shevelev [46] published the book The
Metalanguage of Living Nature.
The beginning of the 21st century has been characterized by increasing
scientific activity by the Slavic Golden Group. In 2001, Alexey Stakhov and
Anna Sluchenkova created a website Museum of Harmony and the Golden
Section www.goldenmuseum.com. The uniqueness of this site in its bilingual
ism (Russian/English) and big concentration of exclusive information about
Golden Section and its applications.
In 2003, the Slavic Golden Group organized the international conference Prob
lems of Harmony, Symmetry and the Golden Section in Nature, Science and Art in the
Ukrainian city of Vinnitsa. Here the great contribution of the Slavic scientists to
the development of the theory of the golden section and its applications was recog
nized in the 38 books written on the golden section. According to the Conference
resolution, the Slavic Golden Group was transformed into the International Club of
the Golden Section. This group stimulated Slavic researchers to further develop their
creative work. Included amongst the many interesting books written by this presti
gious group of leading Slavic scientists, were the following very popular ones:
1. Bodnar, O.J. The Golden Section and NonEuclidean Ggeometry in Sci
ence and Art (2005) (in Ukrainian) [52]. This book is the second edition
of Bodnar’s preceding book [37].
Alexey Stakhov
THE MATHEMATICS OF HARMONY
136
2. Petrunenko, V.V. The Golden Section in Quantum States and its Astro
nomical and Physical Manifestations (2005) (in Russian) [53].
3. Soroko, E. M. The Golden Section, Processes of Selforganization and
Evolution of System. Introduction into General Theory of System Harmony
(2006) (in Russian) [[56]. This book is the second edition of Soroko’s
preceding book [25].
4. Stakhov, A.P., Sluchenkova, A.A., and Scherbakov, I.G. The da Vinci
Code and Fibonacci Series (2006) (in Russian) [55].
Together with the American Fibonacci Association, the Slavic Golden
Group greatly influenced the development of contemporary research in the
field of the golden section and Fibonacci numbers, and their applications.
2.20. Conclusion
The Fibonacci and Lucas numbers are two remarkable numerical sequences,
which are becoming widely known in modern science. Fibonacci numbers were
introduced into mathematics in the 13th century by the famous Italian mathema
tician Leonardo of Pisa (Fibonacci) as the solution to the Fibonacci rabbit repro
duction problem. Fibonacci and Lucas numbers have many interesting mathe
matical properties. If we take the ratio of two adjacent Fibonacci numbers
Fn/Fn1 and direct n towards infinity, the ratio approaches the golden mean in the
limit. The discovery of this mathematical property is attributed to Johannes Ke
pler. In the 17th century Kepler’s contemporary, the great astronomer Giovanni
Cassini, proved the remarkable Cassini formula connecting three neighboring
Fibonacci numbers. If we calculate the numerological values of all the terms of
the Fibonacci series, then we will find that there is an intriguing periodicity, equal
to 24, in this series of numerological values. The French 19th century mathemati
cians Lucas and Binet made great contributions to the development of Fibonacci
number theory. Lucas introduced Lucas numbers into mathematics and Binet
derived the famous mathematical formulas (Binet formulas) which connect Fi
bonacci and Lucas numbers with the golden mean. The Fibonacci and Lucas num
bers show themselves throughout Nature’s structures. The botanic phenomenon
of phyllotaxis is the best recognized amongst them. However, during the last few
decades, Fibonacci numbers have been revealed in the genetic code (Fibonacci
Resonances), in psychology (Lefevre’s experiments), and the extension of market
processes (Elliott Waves), and numerous other areas and disciplines.
Chapter 3
Regular Polyhedrons
137
Chapter 3
Regular Polyhedrons
3.1. Platonic Solids
3.1.1. Regular Polygons
People show an interest in regular polygons and polyhedrons (i.e. polyhedra)
throughout life – from the twoyearold child, playing with wooden cubes, to the
mature mathematician. Some regular and semiregular solids appear in Nature in
the form of crystals, others as viruses seen only by using an electron microscope.
What is a polygon? Recall that geometry may be defined as a science of
space and spatial figures – twodimensional and threedimensional. A two
dimensional geometric figure can be defined as a set of line segments limiting
some part of a plane. Such a planar figure is called a Polygon.
Scientists have been interested for some time in the Ideal or Regular poly
gons, that is, the polygons having equal sides and equal angles. The idea of
“regularity” and “ideal geometric figures” is very old in geometry. It dates back
to the ancient Greek mathematicians and philosophers.
The equilateral triangle is considered to be the simplest regular polygon be
cause it has the least number of sides necessary to limit part of a plane. The square
(four sides), pentagon (five sides), hexagon (six sides), octagon (eight sides), deca
gon (ten sides) and so on to
gether with the equilateral tri
angle (Fig. 3.1) provide a gen
eral picture of the regular poly Equalateral
Cube
Pentagon
Hexagon
gons. It is obvious that there are
no theoretical restrictions on
the number of sides of a regular
Octagon
Nonagon
Decagon
polygon, meaning there are an Heptagon
Figure 3.1. Regular convex polygons
infinite number of polygons.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
138
Note that all regular polygons in Fig. 3.1 are Convex. What is a convex geo
metric figure? The Convex geometric figure is the opposite of a Concave geo
metric figure. To understand the distinction between convex and concave
geometric figures, we examine two famous figures, the pentagon (Fig. 3.2a)
and pentagram (Fig. 3.2b).
The regular pentagon in Fig. 3.2a is a planar
convex figure, while the pentagram in Fig. 3.2b
is a planar concave figure.
The notion of convex and concave geometric
a)
b)
figures should be intuitively clear. Below we clar
Figure 3.2. Regular pentagon (a)
ify these geometric concepts in greater detail when is a convex figure, pentagram (b)
is a concave figure
we begin the study of regular polyhedra.
The first textbook of geometry, Euclid’s Elements, assumed convexity with
out providing a precise definition. Convex polygons have an interesting prop
erty connected with the interior angles of polygons. We know that the three
angles in any triangle always total 180°. As the equilateral triangles have equal
angles, each of its angles must be equal to 60° (60+60+60=180). The angles of
any quadrangular polygon add to 360°. As the angles of a regular polygon are
equal, each angle of the regular quadrangular polygon (square) is equal to 90°
(90+90+90+90=360). The angles of any fivesided polygon add to 540°. It is
clear that each angle of the regular pentagonal polygon (pentagon) is equal to
108° (108+108+108+108+108=540).
How can we calculate the interior angle of any regular ngon? Take any vertex
of a regular ngon and draw from this vertex all possible diagonals within the poly
gon. This process divides the regular ngon into n–2 (n minus 2) nonintersecting
triangles. As three angles of a triangle total 180°, then the interior angles of any
regular ngon total [180° (n2 )], a characteristic property of the convex polygon.
Table 3.1 lists the numerical values for the interior angles of regular ngons.
Table 3.1. Interior angle measures in regular polygons
Name
Triangle
Square
Pentagon
Hexagon
Octagon
Nonagon
Decagon
Dodecagon
...
ngon
Number of sides
3
4
5
6
8
9
10
12
...
n
Sum of interior angles
180
360
540
720
1080
1260
1440
1800
...
180(n2)
Interior angle
60
90
108
120
135
140
144
150
...
180(n2) /n
Chapter 3
Regular Polyhedrons
139
Note that regular polygons have many lines of symmetry. This character
istic is very important in their ability to build up different tessellations. We
can see from Fig. 3.3 that the regular oc
tagon has eight axes of symmetry: one be
tween each pair of opposite vertices and
one between each pair of opposite sides.
The regular pentagon has five axes of
symmetry: one between each vertex and
its opposite side. A regular polygon with Figure 3.3. Symmetry lines of the regular
octagon and the regular pentagon
n sides has n axes of symmetry.
3.1.2. Regular Polyhedra
A polyhedron is a “solid” threedimensional figure similar to twodimen
sional polygons discussed above. Polyhedra have vertices, edges and faces. If a
polyhedron has faces that are regular polygons and if at each vertex exactly the
same number of faces meets, such a polyhedron is called a Regular Polyhedron.
How many regular polyhedra exist? At first sight, the answer to this question is
very simple: as many as there are regular polygons that are faces of regular poly
hedra. However, this is not the case. To find the correct answer to this question
we must divide all regular polyhedra into convex and concave types.
Let us start with the convex polyhedra. Euclid’s Elements give a strict
proof of the fact that only five convex regular polyhedra exist, and their faces
must be one of only three types of the regular polygons: triangles, squares and
pentagons (Fig. 3.4).
a)
b)
c)
d)
e)
Figure 3.4. Platonic Solids: (a) tetrahedron (Fire), (b) octahedron (Air), (c) hexahedron
or cube (Earth), (d) icosahedron (Water), (e) dodecahedron (Universal Mind)
Many books are devoted to the theory of polyhedra. Polyhedron Models
by English mathematician M. Wenninger is the best known amongst them.
The Russian translation of this book was published in 1974 [147]. Bertrand
Russell’s statement was chosen as an epigraph to this book: “Mathematics
possesses not only truth, but also supreme beauty … sublimely pure, and capa
bly of a stern perfection such as only the greatest art can show.”
Alexey Stakhov
THE MATHEMATICS OF HARMONY
140
The book [147] begins with a description of the convex regular polyhedra, that is,
the polyhedra that are regular polygons of one type. These polyhedra are named the
Platonic Solids in honor of the ancient Greek philosopher Plato, who used regular
polyhedra in his cosmology. We begin our consideration with the regular polyhedra
possessing equilateral triangles as faces. The Tetrahedron is the first and simplest of
them (Fig. 3.4a). The key observation is that the sum of the interior angles of the
polygons that are meeting at every vertex is always less than 360 degrees. In the tet
rahedron, three equilateral triangles (the sum of their interior angles is equal to
180°=3×60°) meet at each vertex; thus, their bases create a new equilateral triangle.
The tetrahedron has the least number of faces among the Platonic Solids and there
fore it is the threedimensional analog to the equilateral triangle (which of course has
the least number of sides among the regular polygons).
The Octahedron (Fig. 3.4b) is the next spatial geometric figure based on
equilateral triangles. In the octahedron, four equilateral triangles (the sum of
their interior angles is equal to 240°=4×60°) come together at one vertex; as a
result, a pyramid with a quadrangular base arises. If one connects two such
pyramids by their bases, then the symmetric figure with eight triangular fac
es, called the Octahedron, appears.
Now, we can try to connect 5 equilateral triangles at one vertex (the sum
of the interior angles is equal to 300°=5×60°). As a result, we obtain a spatial
geometric figure with 20 triangular faces called the Icosahedron (Fig. 3.4d).
A square is the next regular polygon (with interior angle of 90°). If we
unite 3 squares at one vertex (the sum of their interior angles equaling
270°=3×90°) and then add to this figure three new squares, we obtain a per
fect geometric figure with 6 faces called a Hexahedron or Cube (Fig. 3.4c).
Finally, there is one more regular polyhedron to construct based on the
use of a pentagon with the interior angle 108°. If we collect 12 pentagons so
that 3 regular pentagons come together at each vertex (the sum of their inte
rior angles is equal to 324°=3×108°), then we obtain the next Platonic Solid
with 12 pentagonal faces called the Dodecahedron (Fig. 3.4e).
The hexagon is the next regular polygon after the pentagon. It has an
interior angle 120°. If we connect 3 hexagons at one vertex, we obtain a sur
face because the sum of their interior angles is equal to 360°=3×120°. This
means that it is impossible to construct a threedimensional geometric fig
ure from hexagons alone. Other regular polygons after the hexagon have
interior angles greater than 120°. This means that we cannot construct spa
tial geometric figures from them. It follows from this examination that there
are only 5 convex regular polyhedra, the faces of which are limited to
equilateral triangles, squares and pentagons.
Chapter 3
Regular Polyhedrons
141
There are surprising geometrical connections between all regular polyhe
dra. For example, the cube (Fig. 3.4c) and the octahedron (Fig. 3.4b) are dual
to one another, that is, they can be obtained from each other, if the centroids of
the faces of the first figure are taken as the vertices of the other and conversely.
Similarly, the icosahedron (Fig. 3.4d) is dual to the dodecahedron (Fig. 3.4e).
The tetrahedron (Fig. 3.4a) is dual to itself. The fact of the existence of only
five convex regular polyhedra is surprising because the number of regular poly
gons is infinite! This fact was proven in Euclid’s Elements.
3.1.3. Numerical Characteristics of Platonic Solids
Before continuing, let us collect some data about the regular polyhedra. Let
• m be the number of polygons meeting at one vertex,
• n be the number of vertices of each polygon,
• f be the number of faces of a polyhedron,
• e be the number of edges of a polyhedron, and
• v be the number of vertices of a polyhedron.
The values of these numbers for Table 3.2. Numerical characteristics of
regular polyhedra
each of the regular polyhedra are list
ed in Table 3.2.
n m
f
e
v
Further, our aim is to show that for Tetrahedron
3 3
4
6
4
3 4
8 12 6
any pair of numbers n and m the values Octahedron
Icosahedron
3
5
20
30 12
of the other parameters f, e, and v are
Hexahedron
4 3
6 12 8
uniquely determined. As two faces come
Dodecahedron 5 3 12 30 20
together at one edge, we can write:
e=nf / 2.
Next, as m faces come together at each vertex, we can write:
(3.1)
v=nf / m.
(3.2)
It is apparent from Table 3.2 that for all five regular polyhedra we have:
f=2+ev.
The result (3.3) is known as Euler’s Polyhedron Theorem.
(3.3)
3.1.4. The Golden Section in the Dodecahedron and Icosahedron
The dodecahedron and its dual the icosahedron (Fig. 3.4d, e) take up a
special place among the Platonic Solids. First of all, it is necessary to empha
size that the geometry of the dodecahedron and icosahedron is directly con
nected with the golden section. The faces of the dodecahedron (Fig. 3.4e) are
Alexey Stakhov
THE MATHEMATICS OF HARMONY
142
pentagons based on the golden section. If we look carefully at the icosahedron
(Fig. 3.4d), we can see that five triangles come together at each vertex of the
icosahedron; here, their external sides produce a pentagon based on the gold
en section. Already these facts demonstrate that the golden section plays an
essential role in these two Platonic Solids.
A duality of the dodecahedron and the icosahedron is expressed in the fact
that the number of the dodecahedron faces (f=12) is equal to the number of the
icosahedron vertices (v=12) and the number of the icosahedron faces (f=20) is
equal to the number of the dodecahedron vertices (v=20), but they both have
the same number of edges (e=30). There is another numerical characteristic
that unites the dodecahedron and the icosahedron: the number of planar angles
on the surface of both spatial figures is equal to 60. As the number of vertices
(and sides) n of the regular polygons (triangle and pentagon) that are the faces
of the icosahedron and dodecahedron are equal to 3 and 5, respectively, and the
number of faces of the icosahedron and dodecahedron are equal to 20 and 12,
respectively, it follows that the following formula for the number of the planar
angles on the surface of the icosahedron and the dodecahedron are valid:
60=3×20=5×12.
(3.4)
There are also deeper confirmations of the fundamental role of the gold
en section in the icosahedron and the dodecahedron. It is recognized that
these Platonic Solids have three unique spheres. The first sphere (the in
scribed sphere or insphere) is a sphere that is inserted into the Platonic Sol
id and touches the centroids of its faces. Define the radius of this insphere
by Ri. The second or middle sphere (midsphere or intersphere) touches the
centroids of its edges. Define the radius of the midsphere by Rm. The third
(external) sphere or circumsphere is circumscribed around the Platonic Solid
and passes through its vertices. Define this radius by Rc. It was proven in
geometry that the radius lengths of the indicated spheres for the dodecahe
dron and the icosahedron with sides equal to 1 are expressed by the golden
mean τ (see Table 3.3).
Table 3.3. Connection of the icosahedron and dodecahedron with the golden mean τ
Icosahedron
Dodecahedron
Rc
Rm
Ri
W 3W
W
W
2
2
2 3
W 3
W
2
W
2
2
2 3W
2
2
Chapter 3
Regular Polyhedrons
143
3(3 − τ)
Rc
=
are equal for both the icosa
Ri
τ
hedron and the dodecahedron. Thus, if the dodecahedron and the icosahe
dron have identical inserted spheres, then their described spheres are also iden
tical. A proof of this mathematical theorem is given in Euclid’s Elements.
Note that the ratios of the radii
Table 3.4. The golden mean in the areas and volumes of the icosahedron and dodecahedron
Icosahedron
Outer area
Volume
Dodecahedron
15W
5 3
5W
6
5
3W
5W
3
2(3 W)
There are other wellknown geometric relations for the dodecahedron and
the icosahedron verifying their connection with the golden mean. For exam
ple, if we take the icosahedron and the dodecahedron with edge length equal
to 1 and compute their external areas and volumes, then they will be expressed
in terms of the golden mean (see Table 3.4).
Thus, there are a large number of relations obtained by the mathemati
cians of antiquity, which verify the remarkable fact, that the golden mean is
the main proportion of the dodecahedron and icosahedron. A connection of
the golden mean with dodecahedron and icosahedron is especially interesting
from the point of view of the socalled “dodecahedronicosahedron doctrine”
considered below.
3.1.5. Plato’s Cosmology
We mentioned above that the regular polyhedra were named Platonic Sol
ids because they played a very important part in Plato’s cosmology.
In Plato’s cosmology, the first four polyhedra personified
four “essences” or “elements.” The Tetrahedron symbolized Fire
with its top pointed upwards; the Icosahedron symbolized
Water as the most “fluid” polyhedron; the Cube symbolized
Earth, as the “steadiest” polyhedron; the Octahedron symbol
ized Air presumably the most “aerial” polyhedron. The fifth
polyhedron, the Dodecahedron, symbolized “the real World,”
Plato
“Universal Reason,” and was considered the main geometrical
(427
347 BC)
figure of the Universe or entire Cosmos.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
144
The ancient Greeks considered harmonious relations as a basis of the Uni
verse, therefore the first four “elements” were connected by the following pro
portion: Earth/Water = Air/Fire. According to Plato, the atoms of the “ele
ments” are in perfect consonances similar to four strings of lyre. It is pertinent
to tell that such system of “elements” that includes the four elements, Earth,
Water, Air and Fire, was canonized also by Aristotle. These four “elements”
remained the four fundamental stones of the Universe within many centuries.
Here we can use an analogy with today’s wellknown four states of substance:
rigid, liquid, gaseous and plasma.
Thus, the ancient Greek idea about the Harmony of the Universe was bound
together with its embodiment in the Platonic Solids. Plato’s idea about the role
of the regular polyhedra in the structure of the Universe influenced Euclid in
his Elements. In this famous book, which over the centuries was the unique text
book of geometry, a description of “ideal” lines and figures is given. A straight
line is the most “ideal” line, and also the regular polygons and polyhedra are the
most ideal geometric figures. It is interesting that Euclid’s Elements begins with
the description of an equilateral triangle that is the simplest regular polygon
and ends with a study of the five Platonic Solids. We mentioned above that the
theory of the Platonic Solids was stated in the 13th, that is, final book of Eu
clid’s Elements. That is why, the ancient Greek mathematician Proclus, a com
mentator on Euclid, put forward the interesting hypothesis about the true pur
pose for which Euclid wrote the Elements. In Proclus’ opinion, Euclid wrote his
Elements in order to provide a complete theory of the construction of the “ideal”
geometric figures, in particular, the five Platonic Solids. In passing he gave in
the Elements some advanced achievements of the Greek mathematics necessary
to give a complete theory of the “ideal” geometric figures! Thus, we can consid
er Euclid’s Elements as the first historically geometric theory of the Harmony of
the Universe, based on the “Golden Section” (division in extreme and mean
ratio) and the Platonic Solids!
3.2. Archimedean Solids and Starshaped Regular Polyhedra
3.2.1. Archimedean Solids
There are 13 semiregular convex polyhedra attributed to Archimedes.
This wellknown set of the perfect geometric figures is named Archimedean or
Semiregular Polyhedra.
Chapter 3
Regular Polyhedrons
145
The Archimedean Solids can be divided into several
groups. The first group consists of the five polyhedra that
are formed from the Platonic Solids by means of the trun
cation of their vertices. For the Platonic Solids the trunca
tion can be made in such a manner that the new faces and
the remaining parts of the old faces are regular polygons.
For example, the tetrahedron (Fig. 3.4а) can be truncated
so that its four triangular faces are converted into hexago
Archimedes
nal faces and the four new regular triangular faces are add (287 – 212 BC)
ed to them. In this manner the five Archimedean Solids are obtained: Truncat
ed Tetrahedron, Truncated Hexahedron (Cube), Truncated Octahedron, Trun
cated Icosahedron, and Truncated Dodecahedron (Fig. 3.5).
a)
b)
c)
d)
e)
Figure 3.5. Archimedean solids: (а) truncated tetrahedron, (b) truncated cube, (c) truncat
ed octahedron, (d) truncated dodecahedron, (e) truncated icosahedron
In his Nobel lecture (1996) the American chemist Richard E. Smalley, one of
the authors of the experimental discovery of fullerenes, spoke about Archimedes
(287 212 BC) as the first researcher of truncated polyhedra, in particular, the
Truncated Icosahedron. It is his opinion, however, that Archimedes may have as
cribed to himself this discovery, even though the icosahedron was truncated long
before him. All these solids were described by Archimedes, although, his original
works on this topic were lost and were known only from secondhand sources.
Various scientists gradually rediscovered all these polyhedra during the Renais
sance, and Kepler finally reconstructed the entire set of Archimedes’ polyhedra in
his 1619 book The World Harmony (“Harmonice Mundi”).
How we can construct the Archi
medean truncated icosahedron from the
Platonic icosahedron? The answer to this
question is given in Fig. 3.6. We can see
from Table 3.2 that five faces converge in
Figure 3.6. Construction of the
each of the 12 vertices of the Platonic icosa Archimedean truncated icosahedron
from the Platonic icosahedron
hedron. If we truncate 12 vertices of the
icosahedron by a plane, then 12 new pentagonal faces will be formed. The old
triangular faces are converted into hexagonal faces. 12 new pentagonal faces to
Alexey Stakhov
THE MATHEMATICS OF HARMONY
146
gether with 20 hexagonal faces compose a truncated icosahedron with 32 faces.
Here the edges and vertices are equal to 90 and 60, respectively.
Another group of Archimedean solids consists of two quasiregular polyhe
dra. A polyhedron is called Quasiregular if it consists of two sets of regular
polygons, let’s say, msided and nsid
ed, respectively, and it is constructed so
that each polygon of the first set is com
pletely surrounded by polygons of the
second set. There are two quasiregular
Archimedean solids named Cuboctahe
dron and Icosidodecahedron (Fig. 3.7).
a)
b)
For the cuboctahedron (Fig. 3.7a) m =
Figure 3.7. Cuboctahedron (a) and
3, n = 4, and for the icosidodecahedron
icosidodecahedron (b)
(Fig. 3.7b) m = 3, n = 5.
There are two Archimedean solids called the Rhombicuboctahedron
(Fig. 3.8a) and the Rhombicosidodecahedron (Fig. 3.8b). The rhombicubocta
hedron (Fig. 3.8a) consists of two kinds of faces, squares and triangles. Each square
is surrounded by four squares and each tri
angle is surrounded by three squares.
Some believe that the rhombicosi
dodecahedron (Fig. 3.8b) is the most
attractive among the Archimedean sol
ids. The rhombicosidodecahedron con
sists of pentagons, squares and triangles.
a)
b)
Figure 3.8.
Each pentagon is surrounded by five
Rhombicuboctahedron (a)
squares and there is a triangle surround
and rhombicosidodecahedron (b)
ed by three squares.
Finally, there are two socalled “snubnosed” versions of the Archimedean
solids – one for the hexahedron (cube), Snub Hexahedron (Fig. 3.9a), another –
for the dodecahedron, Snub Dodecahedron (Fig. 3.9b). The snub hexahedron (Fig.
3.9a) consists of six squares surround
ed by triangles. Each square is surround
ed by four triangles. Each triangle ad
joined to a square is surrounded by two
triangles. The same idea underlies a snub
dodecahedron (Fig. 3.9b). It consists of
12 pentagons where each is surrounded
a)
b)
by five squares. The gap between squares
Figure 3.9. Snub hexahedron (a) and
is filled by triangles.
snub dodecahedron (b)
Chapter 3
Regular Polyhedrons
147
3.2.2. Starshaped Regular Polyhedra
In Polyhedron Models by M. Wenninger [147] we find 75 various mod
els of regular and semiregular polyhedra. The Russian mathematician
Ljusternak, who made much of this area, wrote: “A theory of polyhedra,
in particular, the convex polyhedra, is one of the most fascinating chap
ters of geometry.” The development of this theory is connected with the
names of several outstanding scientists. We mentioned Kepler’s contri
bution to the development of the theory of polyhedra. He wrote the etude
About snowflakes where he noted the following: “Among the regular poly
hedra the cube is the very first one, the beginning and the primogenitor
of others, if it is permissible so to say, its spouse is the octahedron be
cause it has as many vertices as the cube has faces.” Kepler published a
full list of the 13 Archimedean solids and gave them the names under
which they are now known.
Also Kepler started to study the socalled Starshaped Polyhedra that un
like Platonic and Archimedean Solids are regular concave polyhedra. In the
beginning of the 19th century the French mathematician and physicist Poin
sot (1777 1859), whose geometric works relate to starshaped regular poly
hedra, followed Kepler’s work and discovered two new starshaped regular
polyhedra. So, thanks to Kepler and Poinsot’s works, four types of the star
shaped regular polyhedra (Fig. 3.10) became known. In 1812, the French math
ematician Cauchy (1789 1857) proved that other starshaped regular poly
hedra do not exist.
Many readers may ask a question: “For what purpose is it necessary to
study regular polyhedra? What benefit can we have from them?” We could
answer this question by another question: “What benefit can we derive from
music or poetry? Is all that is beautiful merely useful?” The models of polyhe
dra presented in Figs. 3.43.10, first of all, make an aesthetic impression upon
us and can be used
as decorative orna
ments. However, it
will be shown that a
wide manifestation
of regular polyhedra
in natural structures
caused a great deal
of interest in mod
Figure 3.10. Starshaped regular polyhedra
ern science.
(KeplerPoinsot solids)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
148
3.3. A Mystery of the Egyptian Calendar
3.3.1. What is a Calendar?
A Russian proverb states: “Time is the eye of history.” Everything that
exists in the Universe: the Sun, the Earth, stars, planets, known and unknown
worlds – they all have spatialtemporal measurement. Time is measured by
observation of the periodically repeating processes of definite duration.
In remote antiquity people noted that the day is always followed by night,
the seasons of the year alternating on a regular basis: after the winter the spring
comes, after the spring the summer, after the summer the autumn…
By searching for the cause of these phenomena, a person paid attention to
the celestial heavenly bodies, the Sun, the Moon, stars, and the strict period
icity of their movement in the firmament. It was the first celestial observa
tions that preceded the origin of astronomy, one of the most ancient sciences.
As the basis of time measurement, the astronomers had used three important
astronomical phenomena: rotation of the Earth around the axis, motion of the
Moon around the Earth and motion of the Earth around the Sun. The various
concepts of time often depended on the physical phenomenon used for time
measurement. Astronomy knows of a stellar time, solar time, local time, zone
time, nuclear time, etc. The Sun, as well as all other heavenly bodies, partici
pates in the motion of the firmament. Except for diurnal motion, the Sun has
socalled annual motion, and the totality of the annual motion of the Sun on
the firmament is an Ecliptic. If, for example, we fix the location of the constel
lations at some definite moment of time and then repeat this observation each
month, we can see different pictures of the palate. A view of the starry sky
changes continuously: each season has its own picture of vesper constella
tions recurring yearly. Therefore, on the expiration of one year the Sun re
turns to its initial location in the starry sky.
To be oriented conveniently in the stellar world, astronomers divided the
entire firmament into 88 constellations. Each of them has its own name. Among
the 88 constellations, those that are on the ecliptic, play a special role. These
constellations have a generalized title – “the zodiac” (from the Greek word
“zoop” – animal), and are widely known throughout the world as symbols
(signs) with a variety of allegorical meanings and calendar systems.
During its apparent movement along the ecliptic, the Sun intersects 13
constellations. However, astronomers found it necessary to divide the Sun’s
Chapter 3
Regular Polyhedrons
149
path into 12 parts (not 13), by uniting the constellations of Scorpions and the
Dragon in one unified constellation under the common name Scorpio.
The special science of Chronology studies the problems of time measure
ment. This science underlies all calendar systems constructed by mankind. In
antiquity the creation of calendars was one of the major problems of astronomy.
What are “calendars” and “calendar systems?” The word “calendar” orig
inated from the Latin word “calendarium” which meant a “debt book”; in An
cient Rome interest was tracked in such books and was to be paid on the first
day of each month, called “calendas.”
The calendar makers paid great attention to periodicities in the motion of the
Sun, the Moon, as well as, Jupiter and Saturn, the two gigantic planets of the
Solar system. The idea of Jupiter’s calendar with the celestial symbolism of a 12
year animal cycle may be connected to the rotation of Jupiter, which makes its
full rotation around the Sun in approximately 12 years (11.862 years). On the
other hand, Saturn, the second gigantic planet of the Solar system, makes its full
rotation around the Sun in approximately 30 years (29.458 years). By wishing to
coordinate the cyclic movements of the gigantic planets, the ancient Chinese in
troduced the idea of a 60year cycle of the Solar system. During this cycle, Saturn
makes 2 full rotations around the Sun, and Jupiter makes 5 rotations.
From antiquity the calendar makers of Eastern and SouthEast Asia made
use of the following astronomical phenomena: alternation of day and night,
change of lunar phases and alternation of seasons. The use of different astro
nomical phenomena resulted in the creation of three kinds of calendars: the
lunar one based on the motion of the Moon, the solar one based on the motion
of the Sun, and the combination lunarsolar one.
3.3.2. Structure of the Egyptian Calendar
The Egyptian calendar that was produced in the 4th millennium BC was
one of the first solar calendars. One year in this calendar consisted of 365
days. One Year was divided into 12 months, each Month consisted of 30 days;
at the end of the year the 5 holidays that were not part of the month structure
were added. Thus, the Egyptian calendar year had the following structure:
365 = 12×30 + 5. It is important to note that the Egyptian calendar is a prede
cessor of the contemporary calendar.
Why the Egyptians divided the calendar year into 12 months? We know
that there were calendars with other numbers of months in the year. For ex
ample, in the Maya calendar one year consisted of 18 months, each month
being 20 days. Furthermore, why did each month in the Egyptian calendar
Alexey Stakhov
THE MATHEMATICS OF HARMONY
150
have exactly 30 days? One may ask questions concerning the Egyptian sys
tem of time measurement, in particular, regarding the choice of such units of
time, such as Hour, Minute and Second. In particular, the question is: why was
unit of hour chosen so that 1 day=24(2×12) hours? Further, why 1 hour=60
minutes, and 1 minute=60 seconds? The same questions concerning the choice
of the measurement units of angular magnitudes, in particular, why is a cir
cumference divided into 360° that is, why 2π=360°=12×30°? We can ask oth
er questions, for example: why do astronomers find it expedient to introduce
12 “zodiacal” constellations, instead of the 13 the Sun appears to intersect
during its motion along the ecliptic? And why does the Babylonian number
system have a rather esoteric radix, the number 60?
3.3.3. Connection of the Egyptian Calendar with the Numerical
Characteristics of the Dodecahedron
Analyzing the above questions we discover with surprising consistency
the following four numbers are repeated: 12, 30, 60 and the derivative num
ber 360 (360=12×30). Is there some scientific fact that could give a simple
and logical explanation of the use of these numbers in the Egyptian calen
dar, and their system of time and angle measurement? To answer this ques
tion, we return once again to the regular dodecahedron based on the golden
section (Fig. 3.4e).
Did the Egyptians know the dodecahedron? Historians of mathematics
recognize that the ancient Egyptians had information about the regular poly
hedra. The ancient Greek mathematician Proclus attributes to Pythagoras
the construction of all 5 regular polyhedra. However, we know that Pythago
ras borrowed from the ancient Egyptians many mathematical theorems and
discoveries, in particular, the Pythagorean Theorem. Some accounts claim
Pythagoras spent 22 years in Egypt and 12 years in Babylon. Therefore, it is
possible that Pythagoras could have also borrowed the knowledge about the
regular polyhedra from the ancient Egyptians.
However, there is more substantial proof that the Egyptians possessed
information about all 5 regular polyhedra. In particular, the British Muse
um has preserved the dice from Ptolemy’s epoch that have the form of an
icosahedron, the Platonic dual to the dodecahedron. These facts allow us to
put forward the hypothesis that the dodecahedron was known to the Egyp
tians. The very unusual theory of the origin of the Egyptian calendar, as
well as the Egyptian measurement system of the time and geometric angles
follows from this hypothesis.
Chapter 3
Regular Polyhedrons
151
Earlier we found that the dodecahedron has 12 faces, 30 edges and 60
planar angles on its surface. If we accept the hypothesis that the ancient Egyp
tians knew the dodecahedron and its numerical parameters 12, 30 and 60, then
the scientists of antiquity should not have been surprised, when they discov
ered that the cycles of the Solar system are expressed by the same numbers
(12year cycle of Jupiter, 30year cycle of Saturn, and 60year cycle of the
Solar system). Thus, there is a deep mathematical connection between the
Solar system and this perfect spatial figure, the dodecahedron. Scientist of
antiquity apparently came to this conclusion. This may explain why the Egyp
tians (and Plato) chose the dodecahedron as the “Main Geometric Figure,”
that symbolizes the “Harmony of the Universe.” It appears that the Egyptians
made all their main systems (calendar system, systems of time and angle mea
surement) correspond to the numerical parameters of the dodecahedron! Ac
cording to ancient thought, the motion of the Sun on the ecliptic was strictly
circular. By then choosing the 12 Zodiac constellations with the distance of
30°, the Egyptians were able to coordinate the yearly motion of the Sun on
the ecliptic with the structure of their calendar year: one month correspond
ed to the apparent motion of the Sun along the ecliptic between two adja
cent Zodiacal constellations! Moreover, the movement of the Sun one de
gree along the ecliptic corresponded to one day in the Egyptian calendar!
Thus, the ecliptic was divided automatically into 360°. By dividing one day
into two parts, the Egyptians thereby automatically divided each half of one
day into 12 parts (12 faces of the dodecahedron) and introduced the Hour, a
major unit of time. By dividing one hour into 60 minutes (60 planar angles on
the surface of the dodecahedron), the Egyptians introduced the Minute, the
next important unit of a time. And of course this allowed them to introduce
the Second (1 minute = 60 seconds), the smallest unit of time in that period.
Thus, by choosing the dodecahedron as the Main Harmonic Figure of
the Universe and by following strictly to its numerical characteristics (12, 30
and 60), the Egyptians designed a perfect calendar together with the systems
of time and angle measurement that have stood the test over several millen
nia. These systems of course correspond to the golden mean “Theory of Har
mony,” the underlying proportional basis of the dodecahedron.
These surprising conclusions follow from a simple comparison of the dodeca
hedron with the Solar system. And if our hypothesis is correct (let somebody
attempt to deny it), it follows that for several millennia mankind has lived under
the standard of the golden section! And each time, when we look at the index dial
of our watch based on the numerical parameters of the dodecahedron 12, 30 and
60, we touch the “Main Secret of the Universe,” the “Golden Section!”
Alexey Stakhov
THE MATHEMATICS OF HARMONY
152
3.3.4. The Mayan Calendar
The Maya were a very advanced race that made great achievements in the
fields of mathematics and astronomy. Their calendar was, at the time, the most
accurate calendar in the world, surpassed later only by the current Gregorian
calendar (introduced in the 16th century). Their calendar was the product of
the obsession with measuring long periods of time, which they tracked by means
of highly accurate observations of the stars and the planets. The Mayan calen
dar had the following structure: 1 year=360+5=20×18+5. Unlike the Egyptian
calendar, the Mayan calendar year was divided into 18 months of 20 days. It is
clear that the structure of the Mayan calendar was similar to the structure of
the Egyptian solar calendar: 1 year=360+5=12×30+5. Remember that the num
bers 12 and 30 are numerical parameters of the dodecahedron. However, what
does the number 20 refer to in the Maya calendar? Let us address again the
icosahedron and the dodecahedron. In these “sacred” figures, there is one more
“sacred” numerical parameter: the number of the icosahedron faces is 20 and
the number of the dodecahedron vertices is 20! Thus, the Maya undoubtedly
used these numerical characteristics of the icosahedron and dodecahedron in
their calendar by means of the division of one year into 20 months.
3.4. A DodecahedronIcosahedron Doctrine
3.4.1. Sources of the Doctrine
Plato’s cosmology became a source of the socalled DodecahedronIcosa
hedron Doctrine, which by a golden thread passes through all human science.
The essence of this doctrine consists in the fact that the dodecahedron and
the icosahedron are typical forms of Nature in all its manifestations from the
macrocosm down to the microcosm.
According to the remark of one commentator on Plato’s works, for Plato
“all cosmic proportionality is based on the principle of the “golden” or har
monic proportion.” Plato’s cosmology is based on the Platonic Solids. Each
Platonic Solid symbolized one of the five “beginnings” or “elements”: the Tet
rahedron – the body of Fire, the Octahedron – the body of Air, the Hexahedron
(Cube) – the body of Earth, the Icosahedron – the body of Water, the Dodeca
hedron – the body of the Universe. A representation of the general harmony of
the Universe was invariably associated with its embodiment in these regular
Chapter 3
Regular Polyhedrons
153
polyhedra. The fact that the dodecahedron, the “Main Cosmic Figure,” was
based on the golden section gave to the latter a deep sense in its qualifying to
be the “Main Proportion of the Universe.”
We mentioned above that, according to Proclus, the commentator on Eu
clid’s Elements, Euclid did consider a theory of geometric construction of the
Platonic Solids, the “Main Geometric Figures of the Universe,” as the main
goal of Euclid’s Elements. Therefore, Euclid placed this major mathematical
information in the final, that is, 13th book of his Elements.
3.4.2. A Shape of the Earth
The question about Earth’s shape has perennially attracted the attention of
scientists since antiquity. When the hypothesis about the spherical shape of the
Earth received scientific substantiation, the idea arose that the Earth is a dodeca
hedron in shape. Socrates wrote: “The Earth, if to look at it from outside, is sim
ilar to the ball consisting of 12 pieces of skin.” Socrates’ hypothesis found further
scientific development in the works of physicists, mathematicians and geologists.
So, the French geologist de Bimon and the French mathematician Poincare′ pro
posed the hypothesis that the Earth is a deformed dodecahedron in shape.
In the first half of the 20th century, the Russian geologist Kislitsin pro
posed the hypothesis that 400500 million years ago the geosphere’s dodeca
hedral shape was converted into the geoicosahedron. However, such trans
formation appeared incomplete. As a result, the geododecahedron appeared
as if to be inserted into the frame of the icosahedron.
Recently, the Moscow researchers Makarov and Morozov have proposed
another interesting hypothesis that concerns the shape of the Earth. They
supposed that the kernel of the Earth has the shape and properties of a grow
ing crystal that is either a dodecahedron or icosahedron. This crystal influ
ences the development of all natural processes that occur on our planet. Its
energetic field is the cause of this dodecahedralicosahedral structure of the
Earth. The influence of this energetic field reveals itself in the fact that we
find evidence on the Earth’s surface of the dodecahedron and the icosahedron
as if they were inserted into the globe.
In recent years, the hypothesis about the icosahedrondodecahedron shape
of the Earth was subjected to verification. For this purpose, scientists com
bined the axis of the dodecahedron with the axis of the Globe; then they start
ed to rotate the dodecahedron around this axis. They saw that the edges of
the dodecahedron coincide with the gigantic disturbances of Earth’s crust.
Then they started to rotate the icosahedron around the Globe. They saw that
Alexey Stakhov
THE MATHEMATICS OF HARMONY
154
its edges coincide with the smallersized partitioning of Earth’s crust (moun
tain ranges, breaks etc.). These observations support the hypothesis of the
closeness of the tectonic framework of Earth’s crust with the shapes of the
dodecahedron and the icosahedron.
It is as if the vertices of the hypothetical geocrystal are the centers of cer
tain anomalies on the planet: in them all global centers of extreme atmospheric
pressure including the regions of hurricane origin are located. In one of the hubs
of the geoicosahedron (in Gabon) the natural nuclear reactor, which acted 1.7
million years ago, was found. The gigantic mineral fields (for example, Tyumen’s
oil field), the anomalies of the animal world (the lake of Baikal), the centers of
the development of mankind’s civilizations (Ancient Egypt, Northern Mongo
lia, etc.) coincide with many of the vertices of the two polyhedra. All of these
examples tend to confirm Plato’s surprising intuition.
3.5. Johannes Kepler: from “Mysterium” to “Harmony”
3.5.1. “Mysterium Cosmographicum”
Among the fathers of the new European science there was not a person
more mysterious than Johannes Kepler: it seems that he connected two scien
tific epochs not only by his “elliptic” astronomical works, but also his unique
personality. On the one hand, Kepler was a professional astrologer, a dreamer
and visionary, whose style of thinking was unacceptable to the creators of clas
sical science, including Galileo and Newton. On the other hand, this astrolo
ger, almost medieval in his style of thinking, introduced basic concepts into
science. The modern, that is, mechanistic concept of Force was introduced by
Kepler. He introduced the concept of Inertia that distinguishes modern phys
ics from all former physics. At the same time he introduced the concept of
Energy. However, the discovery of New Quantitative Laws of Astronomy was
his main scientific achievement. Kepler is the founder of physics of the heav
ens. This remarkable wordcombination is a subtitle of his basic work: “New
Astronomy based on the Causes, or Physics of the Heavens.”
Johannes Kepler is known by all of educated mankind as the author of
three famous astronomical laws that overturned astronomical ideas that had
existed from antiquity. However, it is less wellknown that these laws were
obtained by Kepler as insights resulting from his ambitious research program
into Universal Harmony that he pursued at a young age.
Chapter 3
Regular Polyhedrons
155
Johannes Kepler was born in 1571 into a poor Protestant family. In 1591,
he enrolled in the Tubingen Academy where he received quite good mathe
matical education. In the Tubingen Academy the future great astronomer be
came acquainted with the heliocentric system of Nicolaus Copernicus. After
graduation from the Academy, Kepler obtained a Masters degree and then
was appointed mathematics professor at the Graz secondary school (Austria).
The small book with the intriguing title Mysterium Cosmographicum, his first
astronomical work, was published by Kepler in 1596 at the age of 25.
Reading the Mysterium Cosmographicum, it is impos
sible not to be surprised by his visionary insights. A deep
belief in the existence of the Harmony of the Universe was
Kepler’s main idea.
Kepler formulated the purpose of his research in the
Foreword as follows:
“To you kind reader! In this book I intend to demonstrate
that our almighty God at the creation of our moving world
and at the disposition of the celestial orbits used the five reg Johannes Kepler
ular polyhedra that are from Pythagoras’ and Plato’s times
(15711630)
and up to now have received great honor. He chose the number and proportions
of the celestial orbits and also the relations between the planetary motions pursu
ant to the nature of the regular polyhedra. I am especially interested in the nature
of three things: why are the planets arranged this way and not otherwise, namely,
the number, sizes and motions of the celestial orbits.”
So, already in the Foreword of his first book the 25yearold Kepler put
forward the main problem of contemporary physics, the problem of the natural
causes of physical phenomena. Though natural today, this problem in Kepler’s
times sounded unusual. In Ptolemy’s and even in Copernicus’ astronomy, this
problem had not been formulated. According to that old tradition, the astrono
mers considered a problem of their science only in terms of a precise description
of planetary motion and the possible prediction of celestial phenomena.
3.5.2. Kepler’s Cosmic Cup
How did Kepler answer the surprising questions raised by him in Mysteri
um Cosmographicum? After the verification of the numerous hypotheses con
nected with the arrangement of planets, Kepler found the following geometri
cal model of the Solar system based on the “Platonic solids” (see Fig. 3.11):
“Earth’s orbit is the measure of all orbits. We place the dodecahedron
around this orbit. The orbit around the dodecahedron is Mars’ orbit. We place
Alexey Stakhov
THE MATHEMATICS OF HARMONY
156
the tetrahedron around Mars’ orbit. The orbit around the tetrahedron is Jupi
ter’s orbit. We place the cube around Jupiter’s orbit. The orbit around the
cube is Saturn’s orbit. Then we insert the icosahedron inside Earth’s orbit.
The orbit inside the icosahedron is Venus’ orbit. We insert the octahedron
inside Venus’ orbit. The orbit inside the octahedron is Mercury’s orbit.”
This model gave Kepler an opportunity to collect together all Platonic Solids
so that they could unite the then “known” planetary spheres. There are only 5
Platonic Solids and only 5 interplanetary spaces, and all of them appear to allow
the regular polyhedra to be placed in them. Could this be mere coincidence?
Kepler’s Cosmic Cup (Fig. 3.11) that inserts the
Platonic Solids into the crystalline spheres embod
ies this model of reality. The most precious trea
sure of ancient geometry, the Platonic Solids, were
used in Kepler’s astronomy. After that Kepler had
the right to say that he comprehended the Universe
as if he created it with his own hands.
Kepler sent his Mysterium Cosmographicum to
Galileo and Brahe and received reassuring respons
es from them. Armed with this support, Kepler ad
dressed the court of Wurttemberg’s duke Freder
Figure 3.11. Kepler’s Cosmic
Cup as a model of the Solar
ick in hopes of receiving the means for the creation
system
of the new model of the Universe, the Cosmic Cup,
in silver. On his application the duke recommended that he first make it in cop
per. The astronomer began to glue a paper model and then threw out the result
after one week of hard work.
Certainly, the creation of the Cosmic Cup was a great success for the young
astronomer. This scientific result had brought scientific renown to Kepler. How
ever, this success was incomplete and even somewhat doubtful, because Kepler’s
basic scientific purpose had not been achieved. In his first scientific work, Kepler
merely anticipated the secret of the world. The Cosmic Cup provided him with
access into the invaluable observational data collected in the Heavenly Castle of
Tycho Brahe. Kepler assisted the great astronomer in the study of Mars. By using
Brahe’s data, the most exact in the world, he intended to polish his new Cosmos,
faceted by Platonic Solids, into a cosmic brilliance. It was necessary only to deter
mine how the planetary orbits could be placed into the Cosmic Cup.
The Cosmic Cup resulted in an important conclusion that disclosed the main
secret of the Universe: the Universe is created on the basis of a general geomet
ric principle, the regular polyhedra! Unfortunately, Kepler’s joy was premature.
In spite of his overfervid character, Kepler had all the characteristics of a seri
Chapter 3
Regular Polyhedrons
157
ous scientist. He believed that the theory should fit the observational data. By
restraining his delight, Kepler undertook to verify his model.
The universal geometric principle allowed Kepler to give answers to two of
the three problems that had been raised by him: (1) to explain the number of
the then known planets (with the help of the five Platonic Solids, it is possible
to construct 6 orbits; the conclusion follows regarding the existence of 6 planets
known in that period); (2) to give an answer about the distances between plan
ets. The answer to the third problem (about the motion of planets) turned out
to be the most difficult and Kepler only attained this answer many years later.
3.5.3. Discovery of Kepler’s First Two Astronomical Laws
Kepler’s model in Fig. 3.11 was based upon the supposition that nature’s
planetary motion is spherical. By obtaining data from the perennial observa
tions of the famous astronomer Brahe, and then carrying out his own observa
tions, Kepler decided to reject the astronomical models of his forerunners,
Ptolemy and Copernicus, and even his own models. After a thorough study of
the planetary orbits, he came to the following conclusion:
“The fact that planetary motions are circular is confirmed by their inces
sant recurrence. The human intellect, by extracting this truth from experi
ence, at once concludes from here, that the planets are rotated on the ‘ideal’
circles, because among the planar figures a circle and among the spatial fig
ures a sphere can be considered as the most perfect geometric figures. Howev
er, after closer examination we can conclude that experience gives another
result, namely, the planetary orbits differ from simple circles.”
The results of this work are presented in Kepler’s main book A New As
tronomy Based on the Causes, or Physics of the Heavens published in 1606. The
importance of this book consists, first of all, of the fact that Kepler therein
formulates the first two of three astronomical laws named after him. Accord
ing to Kepler’s First Law, the orbit of each planet is an Ellipse with the Sun as
one of its foci. Kepler’s Second Law asserts that the areas, covered by the radii
vectors of the planets, are constant.
Kepler’s Laws are the first quantitative laws that are of great importance
for the development of astronomy. By testing the Cosmic Cup, Kepler found
that this model based upon circular planetary orbits did not fit the experi
mental data. Therefore, Kepler decided to reject the idea of circular planetary
motion. After numerous and difficult calculations Kepler found that the or
bits of planets were not circles but Ellipses. This result dramatically altered
much of the basis of previous cosmology.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
158
3.5.4. “Harmonices Mundi”
Kepler’s third astronomical law, formulated by him in Harmonices Mundi
(The Harmony of the World) completed the creation of a new astronomy.
Kepler’s first two laws could not answer one major question of astronomy.
How is it that the planetary distances from the Sun change? Kepler tried to
grasp a new principle for the solution to this challenge. He used his philo
sophical views going back to Pythagoras and Plato. According to Kepler’s
deep belief, Nature was created by God on the basis of not only mathematical,
but also harmonious principles. Kepler believed in the “music of the spheres”
that generate bewitching melodies embodied not in sounds, but in the move
ments of the planets capable of generating harmonious chords. Based on this
idea, Kepler combined mathematical and musical arguments to discover the
third law of planetary motion that asserts the following:
“If Т is the cycle of time of a planet’s orbit around the Sun, and D is its
middle distance from the Sun, then they are connected by the following cor
relation: T 2=kD3, where k is a constant value equal for all planets.”
Kepler arrived at the following conclusion from this discovery: “Thus, the
heavenly movements are a never ending polyphonic music that is perceived
by the human intellect, but not the ear.”
The Russian scientist Predtechensky (18601904), Kepler’s biographer,
wrote about this Kepler’s discovery as follows: “The wonderful harmony, reign
ing in the world, was perceived by Kepler not only in an abstract sense of the
organization of the Universe. The harmony was sounding in its poetic soul by a
true form of music, which could be understood by us if we could enter into the
circle of his ideas and would be imbued with his mighty enthusiasm about the
marvelous construction of the world and the Pythagorean belief in numerical
relations. Really, it is surprising that what makes sounds ‘beautiful’ for hearing
depends upon a strict numerical ratio, for example, between the lengths of the
strings, which produce the sounds as discovered by Pythagoras. But in Kepler,
undoubtedly, part of Pythagoras’ soul lived on, and it is not accidental that he
saw the numerical ratios in the planetary cosmos.”
Kepler’s third Law is the outstanding scientific result, at the summit of
his career. The result obtained filled Kepler’s soul with great joy and grati
tude for the Creator. He expressed this gratitude in the following words:
“The wisdom of the Creator is endless and his glory and power are bound
less. You, the heavens, glorify and praise be to Him! The Sun, the Moon and
the planets, glorify God with their ineffable language! Celestial harmonies,
comprehended by His wonderful creations, do glorify and praise to Him! Let
Chapter 3
Regular Polyhedrons
159
my soul eulogize the Creator! Everything is built by Him and everything is
embodied within Him. Everything the best which is known to us is created by
Him in our bustling life. Praise, Honor and Glory to Him for all of eternity!”
The dramatic period in astronomical development was completed. This pe
riod ended with the discovery of three major astronomical laws of planetary
motion, Kepler’s Laws. This history started with an original model of the Solar
system based on the Platonic Solids. Though Kepler’s model, in the end, ap
peared somewhat erroneous, Kepler did not reject from this model what he de
scribed in his Mysterium Cosmographicum. He always considered this model to
be one of his greatest scientific achievements. By submitting to the requests of
his friends, Kepler in the twilight of his life decided to undertake the second
main issue of his first book “for benefit not only to booksellers, but also scien
tists.” Addressing the new readers, Kepler wrote in the dedication with pride:
“Almost 25 years have passed since I issued a small book Mysterium Cos
mographicum. Though in that time I was very young and this publication was
my first astronomical work, nevertheless the success accompanying this book
in subsequent years, testifies eloquently that no one could write a first book
with a more substantive, successful and valuable treatment of the subject. It is
as if an oracle from the heavens dictated through me the chapters of this book,
because all of them, as is generally accepted, were excellent and corresponded
to the truth. During the 25 years, the chapters of this book illuminated my
way in astronomy many times. Almost all my astronomical works, which I
published by this time, had their beginning in one or another chapter of my
first book, and therefore these later books can be considered to be a more in
depth or more complete presentation of these original chapters.”
3.5.5. Life through the Centuries
Kepler’s life is an example of scientific selflessness based upon the timeless
belief in the Harmony of the Universe. His entire life was a struggle on two
fronts. On the one hand, the “mathematician of the famous province Stiria” strug
gled against poverty and the almost intolerable “life” of the poor having so many
children. His life was accompanied by the depression of his wife, the untimely
deaths of his children, the charges of sorcery against his mother, and the dull
ness of coreligionists. On the other hand, Kepler’s scientific life was a world of
calculations, improbable in their complexity. “The intense, incessant and vigor
ous thoughts” were the basis of Kepler’s unique scientific life.
However, old age was coming. Kepler’s death (in 1630) interrupted his
work on the last book Somnium (Dreams), a science fiction novel about flight
Alexey Stakhov
THE MATHEMATICS OF HARMONY
160
to the Moon. Unfortunately, the Harmony was not written in this book. There
were no captious verifications; there were no new hypotheses. Kepler was tired:
“My brain got tired when I attempted to understand what I wrote, and it is
already difficult for me to reestablish a connection between figures and text
found by me at an earlier time …”
So, the drama had been finished. With Kepler’s death, his discoveries were
forgotten. Even the wise Descartes did not know about Kepler’s works. Gali
leo had not found necessary to read his books. Only in Newton’s works Ke
pler’s Laws find a new life. However, the Harmony is not of interest to New
ton. He is interested in Equations. A new era had arrived.
Kepler’s life terminated the epoch of “scientific romanticism,” the epoch
of the Harmony of the Golden Section that is characteristic of the Renais
sance. On the other hand, his scientific works became the beginning of the
new science that started with the works of Descartes, Galileo and Newton.
And in conclusion we once again remember Kepler’s wellknown state
ment about the Golden Section:
“Geometry has two great treasures: one is the Theorem of Pythagoras; the
other, the division of a line into extreme and mean ratio. The first we may
compare to a measure of gold; the second we may name a precious jewel.”
For those, who treat skeptically Kepler’s comparison of the Pythagorean
Theorem with the Golden Section, we should remind them that Kepler was
not only a great astronomer, but also a great mathematician! Predictions of
great scientists can sometimes dramatically advance the development of a sci
ence. The development of modern science confirms that Kepler was right: the
Golden Section becomes one of the major ideas of modern science!
Unfortunately, after Kepler’s death, the Golden Section, that was consid
ered by him to be one of the great “treasures of geometry,” was forgotten. This
unfortunate ignorance was continued for almost two centuries. With few ex
ceptions interest in the Golden Section was revived only in the 19th century!
3.6. The Regular Icosahedron as the Main Geometrical Object of Mathematics
3.6.1. Felix Klein
Among the five Platonic Solids, the icosahedron and the dodecahedron
have a special place. In Plato’s cosmology the icosahedron symbolizes Water,
and the dodecahedron the Harmony of the Universe. These two Platonic
Chapter 3
Regular Polyhedrons
161
Solids are connected directly with the pentagon and through it the Golden
Section. The dodecahedron and icosahedron give rise to the socalled “icosa
hedrondodecahedron doctrine” that permeates the history of human culture
from Pythagoras and Plato up to the present. It is not an accident that this
doctrine received a unique boost or development in the works of the German
mathematician Felix Klein (18491925).
Felix Klein, a graduate of the University of Bonn, was born
in 1849 and died in 1925. Beginning in 1875 he worked as a
Professor of the Higher Technical School in Munich, then from
1880 as a Professor of the University of Leipzig. In 1886 he
arrived in Gettingen, where he headed the Mathematical In
stitute of the University of Gettingen. During the first quar
ter of the 20th century this Mathematical Institute was rec
ognized as the World’s leading center of mathematics.
Klein’s main works were dedicated to NonEuclidean
Felix Klein
(18491925)
geometry, and the theories of continuous groups, algebraic
equations, and elliptic and automorphic functions. Klein presented his ideas
in the field of geometry in a Comparative Consideration of the New Geometri
cal Researches (1872) known under the title Erlangen’s Program.
According to Klein, each geometry is some invariant theory of a special
group of transformations. By dilating or narrowing down the group, it is pos
sible to pass from one type of geometry to another. The Euclidean geometry is
a science about invariants of the metric group, projective geometry about
invariants of the projective group, etc. The classification of transformation
groups gives us the classification of the geometries. A proof for the existence
of different NonEuclidean geometries is considered to be Klein’s most essen
tial achievement.
3.6.2. Elementary Mathematics from the Point of View of Higher
Mathematics
Felix Klein was not only a wellknown mathematiciantheorist, but also a
reformer of mathematical education in the schools. Prior to World War I he
organized the commission on reorganization of mathematical teaching. The
book Elementary Mathematics from the Point of View of Higher Mathematics
was devoted to the development of a solution to this problem.
Mathematics in the 19th century produced a number of remarkable ideas
that influenced all branches of knowledge and engineering. The main idea
behind the reformers headed by Klein was to increase the role of mathematics
Alexey Stakhov
THE MATHEMATICS OF HARMONY
162
and natural science in the general education. The study of natural sciences
and mathematics, a deepening of the connection between theoretical and ap
plied mathematics, the introduction of functional analysis and calculus into
mathematical teaching, and a broad use of graphic methods are the main prin
ciples suggested by Klein and his followers as a basis for mathematical educa
tion. It is important to recognize that Klein’s ideas are rather topical for mod
ern mathematical education.
3.6.3. Role of the Icosahedron in Mathematical Progress
Besides the Erlangen Program and other outstanding mathematical
achievements, Klein’s greatness consists of the fact that 100 years ago he pre
dicted an outstanding role for the Platonic Solids, in particular, the icosahe
dron in the future development of science. In 1884, Klein published the book
Lectures on the Icosahedron and Solution of the 5th Degree Equations [58] ded
icated to a geometric theory of the icosahedron.
The icosahedron (and its dual, the dodecahedron) play a special role in
“living” nature; many viruses and other living things have the shape of the
icosahedron, that is, the icosahedron shape and pentagonal symmetry play a
fundamental role in the organization of living substance.
According to Klein, the tissue of mathematics extends widely and freely
by the sheets of the different mathematical theories. However, there are geo
metric objects, in which many mathematical theories converge. Their geome
try unites these mathematical theories and allows us to embrace the general
mathematical sense of the various theories. The icosahedron, in Klein’s opin
ion, is just such a mathematical object. Klein treats the regular icosahedron as
the central mathematical object, from which the branches of the five mathe
matical theories follow, namely: Geometry, Galois’ Theory, Group Theory, The
ory of Invariants and Differential Equations.
Thus, the great mathematician Felix Klein following after Pythagoras, Pla
to, Euclid, Johannes Kepler could realize the fundamental role of the Platonic
Solids, in particular the icosahedron, for the development of science and math
ematics. Klein’s main idea is extremely simple: “Each unique geometrical object
is somehow or other connected to the properties of the regular icosahedron.”
Unfortunately, Klein’s contemporaries could not understand and appre
ciate the revolutionary importance of Klein’s idea proposed by him in the
19th century. However, its significance was appreciated one century later,
when the Israeli scientist Dan Shechtman discovered in 1982 a special alloy
with an icosahedral phase called Quasicrystals. And the famous researchers
Chapter 3
Regular Polyhedrons
163
Robert F. Curl, Harold W. Kroto and Richard E. Smalley discovered in 1985
a special kind of carbon called Fullerenes. In 1996, they became Nobel Prize
winners for their discovery. It is important to emphasize that the quasicrys
tals are based on the Platonic icosahedron and the fullerenes on the
Archimedean truncated icosahedron.
3.7. Regular Polyhedra in Nature and Science
3.7.1. Symmetry Groups of Regular Polyhedra
The influence of Klein’s Erlangen Program on school geometry is especial
ly important. The influence of Klein’s “group approach” can be traced in all
themes of school geometry. Each geometric figure F determines some group of
movements; this group contains all those movements that convert the figure F
in a manner similar to itself. This is called the Symmetry Group of the figure F.
In many respects, knowledge of the symmetry group of the figure F deter
mines the geometric properties of that figure.
Let us consider in greater detail some im
portant concepts of symmetry theory. We start
with the Symmetry Plane P that is familiar to
us from the previous Chapter. We offer to the
reader to be convinced that a square has four
planes of symmetry (4Р), however, a rectangle
has only two planes of symmetry (2Р). It can
be easily proven that a cube has 9 planes of sym
metry (Fig. 3.12), that is, 9Р.
Let us now consider the second type of sym
metry elements, the Axis of Symmetry. The axis
Figure 3.12. The 9 planes of
symmetry of the cube
of symmetry is based upon a straight line,
around which the identical parts of a symmetric figure can be repeated a par
ticular number of times. These identical parts are located so that after a turn
around, the symmetry axis on the certain angle the figure occupies the same
position as before the turn. As a result, the figure comes to “selfcoincidence.”
A faceted glass in a glass holder is the best visual example of “selfcoincidence.”
Taking the glass out of the glass holder and then inserting it back in the changed
position, we execute the operation of “selfcoincidence.” The number of all
Alexey Stakhov
THE MATHEMATICS OF HARMONY
164
possible “selfcoincidences” that can be carried out around a given axis is named
its Symmetry Order. Usually the axis of symmetry is denoted by the capital
letter L, but its order is denoted by a small subscript number that stands after
this capital letter. Thus, for example, the symmetry axis of the 3rd order is
denoted by L3. It is clear, that the equilateral triangle has the symmetry axis
L3, the square – L4, the pentagon – L5, and the circle has the symmetry axis of
the infinite order L∞.
As an example, for demonstration of symmetry axes, we examine a cube
(Fig. 3.13).
a)
b)
c)
Figure 3.13. The symmetry axes of the cube: 3L4(a), 4L3(b), 6L2(c)
We can draw three axes of symmetry of the 4th order through the center of a
cube perpendicularly to each pair of opposite faces (Fig. 3.13a). This means, that a
cube has 3 axes of symmetry of the fourth order, that is, 3L4. A cube has 8 vertices.
We can draw a triple symmetry axis that coincides with the corresponding diagonal
of the cube through each pair of the opposite vertices of the cube (Fig. 3.13b). This
means, that the cube has 4 symmetry axes of the 3rd order, that is, 4L3. The cube has
12 edges. We can draw double symmetry axes through the middles of each pair of
the opposite edges, in parallel to the diagonals of the faces (Fig. 3.13с). This means,
that the cube has 6 symmetry axes of the 2nd order, that is, 6L2. Therefore, the full
set of the cube axes of symmetry is the following: 3L44L36L2.
There are geometric figures that have symmetry axes of the infinite order
L∞ . The socalled “figures of rotation,” cylinder, cone, etc. have similar axes. Any
diameter of a fullsphere is the axis of the type L∞. This means, that a fullsphere
has an infinite set of symmetry axes of infinite order, that is, ∞ L∞.
Now, let us consider one more element of symmetry, the Center of Symmetry.
The center of symmetry is a special point inside the figure, when any straight
line, drawn through this point, meets an identical point of the figure at equal
distances from this point. The cube, for example, has a center of symmetry.
Usually for the description of symmetry of some geometric figure a full set
of symmetries is used. For example, the symmetry group of a snowflake is L66P.
This means that a snowflake has one symmetry axis of the 6th order L6, that is,
Chapter 3
Regular Polyhedrons
165
the snowflake “selfcoincides” 6 times at its rotation around the symmetry axis,
and 6 planes of symmetry. The symmetry group of camomile having 24 petals is
L2424P, that is, the camomile has one symmetry axis of the 24th order and 24
planes of symmetry. By uniting all symmetry elements of the cube, we can write
the following symmetry group of the cube: 3L44L36L29PC.
The cube
Table 3.5. Symmetry groups of the Platonic Solids
is, of course,
Polyhedron
A shape of faces
Symmetry group
one of 5 Pla
4 L3 3 L2 6 P
Tetrahedron
Equilateral triangles
tonic Solids.
3 L4 4 L3 6 L2 9 PC
Cube
Squares
Table 3.5 pre
sents the sym
3 L4 4 L3 6 L2 9 PC
Octahedron
Equilateral triangles
metry groups
6 L5 10 L3 15 L2 15 PC
Dodecahedron
Pentagons
of all Platonic
6 L5 10 L3 15 L2 15 PC
Icosahedron
Equilateral triangles
Solids.
Analysis of the symmetry groups of Platonic Solids, given in Table 3.5,
shows that the symmetry groups of the cube and the octahedron, on the one
hand, and the dodecahedron and the icosahedron, on the other hand, coin
cide. Thus the dodecahedron is dual to the icosahedron, and the cube is dual
to the octahedron.
3.7.2. Applications of Regular Polyhedra in the Living Nature
Ernst Heinrich Philipp August Haeckel (18341919) was the eminent Ger
man biologist and philosopher. He became famous following the publication
of his book Kunstformen der Nature (Artshapes of Nature)
Haeckel wrote in his book: “Nature has an inexhaustible number of sur
prising creations, which by beauty and variety far surpass all works created
by human art.”
Nature’s creations presented in Haeckel’s book are beautiful and sym
metric (see Fig. 3.14). This fact is an inseparable property of natural harmo
ny. In his book he gave the examples of the singlecell organisms similar to
the icosahedron in shape. The icosahedron attracted the attention of biolo
gists in their disputes concerning the shape of viruses. The virus cannot be
absolutely round as was considered earlier. To establish its form, the biolo
gists took the various polyhedra and directed a light on them under the same
angle, such as a stream of atoms on the virus. It is proved that only the icosa
hedron gives a precisely identical shadow. It is considered that the geomet
rical properties of the icosahedron allow it to preserve genetic information
in the best possible manner.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
166
Figure 3.14. Artshapes of Nature from Haeckel’s book Kunstformen der Nature (1904)
The regular polyhedra are the most “preferred” and “symmetric” figures.
Nature makes wide use of this property of the polyhedra. The crystals of many
chemical substances have the shape of regular polyhedra. For example, a crys
tal of table salt NaCl has the shape of a cube, a monocrystal of the potassium
alum has the shape of an octahedron, a crystal of the chemical substance FeS
has the shape of a dodecahedron, a sodium sulphite – a tetrahedron, a boron
Br – an icosahedron and so on.
Chapter 3
Regular Polyhedrons
167
Today it has been proven that the process of the formation of a human
embryo from the ovum is carried out by its divisions according to the “bina
ry” law, that is, at first the ovum is transformed into two cells, these new
cells are transformed in turn into four cells, etc. Earlier we considered that
at the second stage of the division the four cells form a square. In fact, it
occurs in another manner: at the second stage of the division, the four cells
form the tetrahedron (Fig. 3.4а). The cells are further divided into eight;
together forming a geometric figure that consists of one tetrahedron point
ing upward and the oth
er tetrahedron pointing
downwards, connected
together. As a result we
obtain the Star Tetrahe
dron that reminds one of
Egg of life
Tetrahedron
Star Tetrahedron
the Egg of Life in esoter
Figure 3.15. A star tetrahedron
ic sciences (Fig. 3.15).
3.7.3. “Parquet Problem” and Penrose Tiling
Since ancient times, “parquet’s problem” in geometry has been the prob
lem of how to fill a plane with regular polygons. Already the Pythagoreans
proved that only regular (equilateral) triangles (symmetry axis of the 3rd or
der), squares (symmetry axis of the 4th order) and hexagons (symmetry axis
of the 6th order) can be solutions to this problem. “Parquet’s problem” has a
direct relation to the main law of crystallography, according to which only
the symmetry axes of the 3rd, 4th and 6th
1
order are allowed in crystals. The symme
1/τ
36°
try axes of the 5th order and more than 6
are prohibited in crystals.
The English mathematician Sir Roger
а)
Penrose was one of the first scientists, who
1
found another solution to “parquet’s prob
72°
lem.” In 1972, he covered a plane in a non
τ
periodic manner, by using only two simple
polygons. In the simplest form, the Penrose’s
Tiles are a nonrandom set of rhombi of two
types, the first one (Fig. 3.16a) has the in
b)
ternal angle 36°, and the second one (Fig.
Figure 3.16. Penrose tiles: (a) “thin”
3.16b) has the internal angle 72°.
rhombus; (b) “thick” rhombus
Alexey Stakhov
THE MATHEMATICS OF HARMONY
168
To understand the mathematical essence of the Penrose Tiles, we return
to the Pentagon and Pentagram (Fig. 3.17).
The pentagram contains a number of char
A
acteristic isosceles triangles. The triangle of the
type ADC is the first of them. The acute angle at
the vertex of A is 36°, and the ratio of the side
F
G
B AC=AD to the base DC is equal to the golden
E
mean, that is, the given triangle is the golden isos
H
L
celes triangle. Note that the pentagram consists
of 5 identical golden isosceles triangles ADC,
K
BED, CAE, DBA, and ECB (the 5 crossing trian
C
D
gles). Besides, we have in the pentagram
Figure 3.17. Pentagram
5 smaller “golden” isosceles triangles AGF, BHG,
CKH, DLK, and EFL similar to ACD. If we now take two such triangles and
connect them together by their bases, we obtain the Penrose’s tile displayed
in Fig. 3.16a and named “Thin” Rhombus. The “thin” rhombus has four verti
ces with the following angles: 36°, 36°, 144°, 144°.
Now, let us consider one more type of isosceles triangle presented in the
pentagram, for example, ABE. In such triangle the acute angles at the vertices
Е and B are each 36°, and the obtuse angle at the vertex A is 108°. Note that
the ratio of the base ЕВ of the triangle ABE to its sides AE=AB is equal to the
golden mean, that is, this triangle is also the golden isosceles triangle. Note
also, the pentagram has 5 identical golden isosceles triangles of this kind, name
ly, ABE, BCA, CDB, DEC, and EAD. If we connect two such triangles together
at their bases, we obtain the second Penrose tile represented in Fig. 3.16b and
named “Thick” Rhombus. The thick rhombus has the four vertices with the
angles: 72°, 72°, 108°, 108°.
Below in Fig. 3.18 we can see a process of sequential construction of the
Penrose Tiling. Take 5 “thick” rhombi and connect them together, as shown in
Fig. 3.18a. Then, we add to the figure in Fig. 3.18a the “thick” and “thin”
rhombi, as shown in Fig. 3.18b. Figure 3.18c is a further development of the
Penrose tiling.
It is proved that the ra
tio of the number of “thick”
rhombi (Fig. 3.16b) to the
number of “thin” rhombi
(Fig. 3.16a) in the Penrose
a)
b)
c)
tiling aims in the limit for
Figure 3.18. Penrose tiling
the golden mean.
Chapter 3
Regular Polyhedrons
169
3.7.4. Quasicrystals
On November 12, 1984 in a small article, published in
the authoritative journal Physical Review Letters, the ex
perimental proof of the existence of a metal alloy with ex
clusive physical properties was presented. The Israeli physi
cist Dan Shechtman was the author of this article.
Dan Shechtman is Philip Tobias Professor of Materials
Science at the Israel University Technion. A special alloy
discovered by Professor Shechtman in 1982 and called Quasi The Israel physicist
Dan Shechtman
crystal is the focus of his research. By using methods of elec
tronic diffraction, Shechtman found new metallic alloys having all the symptoms
of crystals. Their diffraction pictures were composed from the bright and regular
ly located points similar to crystals. However, this picture is characterized by the
socalled “icosahedral” or “pentagonal” symmetry, strictly prohibited according
to geometric reasons. Such unusual alloys are called “quasicrystals.”
The concept of quasicrystals generalizes and completes the definition of
a crystal. Gratia wrote in the article [148]: “A concept of the quasicrystals is
of fundamental interest, because it extends and completes the definition of
the crystal. A theory, based on this concept, replaces the traditional idea about
the ‘structural unit,’ repeated periodically, with the key concept of the distant
order. This concept resulted in a widening of crystallography and we are only
beginning to study the newly uncovered wealth. Its significance in the world
of crystals can be put at the same level with the introduction of the irrational
to the rational numbers in mathematics.”
What are quasicrystals? What are their properties and how we can de
scribe them? We mentioned above that according to the Main Law of Crystal
lography some strict restrictions are imposed on the structure of a crystal.
According to classical ideas, the crystal is constructed from one single cell.
The identical cells should cover a plane densely without any gaps.
As we know, the dense filling of a plane can be carried out by means of Equi
lateral Triangles (Fig. 3.19а), Squares (Fig. 3.19b) and Hexagons (Fig. 3.19d). A
dense filling of the plane by means of Pentagons is impossible (Fig. 3.19c).
а)
b)
c)
d)
Figure 3.19. A dense filling of a plane can be carried out by means of equilateral triangles (а),
squares (b) and hexagons (d)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
170
Penrose tiling (Fig. 3.18) that is a “planar analogy” of quasicrystals was
used for the theoretical explanation of the quasicrystal phenomenon. In the
spatial model, the “regular icosahedrons” (Fig. 3.20) played the role of “Pen
rose rhombi” in the planar model. By using regular icosahedrons, we can ful
fill a dense filling of threedimensional space.
What is the practical significance of the discovery of
quasicrystals? Gratia wrote in [148] that “the mechanical
strength of the quasicrystals increased sharply; here the
absence of periodicity resulted in slowing down the distri
bution of dislocations in comparison to the traditional
metals .… This property is of great practical significance:
the use of the “icosahedral” phase allows for light and very
stable alloys by means of the inclusion of smallsized frag
Figure 3.20.
A regular icosahedron
ments of quasicrystals into the aluminum matrix.”
What is the significance of the discovery of quasicrystals from the point
of view of the main idea of our book, the golden mean? First of all, this discov
ery is a great triumph for the “icosahedrondodecahedron doctrine,” which
passes throughout the history of natural sciences and is a source of profound
ly practical scientific ideas. Secondly, the quasicrystals shattered the con
ventional picture of an insuperable barrier between the mineral world where
the “pentagonal” symmetry was prohibited, and the living world, where the
“pentagonal” symmetry is widespread.
Note that Dan Shechtman published his first article about the quasicrys
tals in 1984, that is, exactly 100 years after the publication of Felix Klein’s
Lectures on the Icosahedron in 1884. This means that this discovery is a wor
thy gift to the centennial anniversary of Klein’s book, in which the famous
German mathematician predicted an outstanding role for the icosahedron in
future scientific development.
3.7.5. Fullerenes
Fullerenes were an important
modern discovery in chemistry.
This discovery was made in 1985,
several years after the quasicrys
tal discovery. The “fullerene” is
named after Buckminster Fuller
a)
b)
(1895 1983), the American de
Figure 3.21. Fuller’s inventions:
signer, architect, poet, and inven (a) Buckminster Fuller; (b) Fuller’s geodesic dome
Chapter 3
Regular Polyhedrons
171
tor. Fuller created a large number of inventions,
primarily in the fields of design and architecture.
The bestknown invention was the Geodesic
Dome based on the truncated regular icosahedron
(Fig. 3.21b).
The geodesic dome, Montreal Biosphere, de
signed by Buckminster Fuller for the American
Pavilion at Expo 67 (Fig. 3.22), is Fuller’s best
known architectural construction.
Figure 3.22. The Montreal
Fuller was the author of many inventions in Biosphere, formerly the American
Pavilion of Expo 67
designing and architecture. In the USA, a post
age stamp was produced immortalizing Buckminster Fuller’s contributions to
architecture and science (Fig. 3.23). Fuller’s head on this postage stamp de
picts the structure of his geodesic dome.
After the discovery of Fullerenes, the name of Buckminster
Fuller became famous worldwide. The title “fullerenes” refers
to the carbon molecules С60, С70, С76, and С84 in which all the
atoms are on a spherical or spheroid surface. In these molecules
the atoms of carbon are located at the vertices of regular hexa
gons or pentagons that cover the surface of the sphere or spher
oid. We start from a brief history of the C60 molecule. This mol
ecule plays a special role among the fullerenes. It is character
Figure 3.23. The
U.S. postage stamp ized by the greatest symmetry and as a consequence is highly
immortalizing
stable. By its shape, the С60 molecule reminds one of a typical
Buckminster Fuller
white and black soccer ball (Fig. 3.24) that has the structure of
a truncated regular icosahedron (Fig. 3.5e, Fig. 3.6).
The atoms of carbon in this molecule are located on the
spherical surface at the vertices of 20 regular hexagons and
12 regular pentagons; here each hexagon is surrounded by
three hexagons and three pentagons, and each pentagon is
surrounded by five hexagons (Fig. 3.25).
The most striking property of the C60
Figure 3.24.
The Soccer Ball molecule is its high degree of symmetry.
There are 120 symmetry operations that convert the mole
cule into itself making it the most symmetric molecule.
It is not surprising that the shape of the C60 molecule
has attracted the attention of many artists and mathemati Figure 3.25. The
cians over the centuries. As mentioned earlier, the truncat
structure of the
С60 molecule
ed icosahedron was already known to Archimedes. The old
Alexey Stakhov
THE MATHEMATICS OF HARMONY
172
est known image of the truncated icosahedron was found in the Vatican library.
This picture was from a book by the painter and mathematician Piero della
Francesca. We can find the truncated icosahedron in Luca Pacioli’s Divina Pro
portione (1509). Also Johannes Kepler studied the Platonic and Archimedean
Solids actually introducing the name “truncated icosahedron” for this shape.
The fullerenes, in essence, are “manmade” structures following from funda
mental physical research. They were discovered in 1985 by Robert F. Curl, Harold
W. Kroto and Richard E. Smalley. The researchers named the newlydiscovered
chemical structure of carbon C60 the Buckminsterfullerene in honor of Buckmin
ster Fuller. In 1996 they won the Nobel Prize in chemistry for this discovery.
Fullerenes possess unusual chemical and physical properties. At high pressure
the carbon С60 becomes firm, like diamond. Its molecules form a crystal structure as
though consisting of ideally smooth spheres, freely rotating in a cubic lattice. Ow
ing to this property, С60 can be used as firm greasing (dry lubricant). The fullerenes
also possess unique magnetic and superconducting properties.
3.8. Applications of Regular Polyhedra in Art
3.8.1. Leonardo da Vinci’s Methods of Regular Polyhedra Representation
Many authors pay particular attention to Leonardo da Vinci’s original
methods of the spatial representation of icosahedron, dodecahedron and trun
cated icosahedron, for the book Divina Proportione (1509) by his contempo
rary, the Franciscan monk and mathematician Luca Pacioli (1445 1514). It
is probably impossible to consider Leonardo’s participation in studying such
perfect geometrical figures as Platonic and Archimedean Solids as a mere co
incidence. And what is more, this fact is deeply symbolic. A true titan of the
Renaissance, artist, sculptor, scientist, engineer and inventor, Leonardo da
Vinci (1452 1519) is a symbol of the inseparable bond between Art and Sci
ence. His interest in such fine and highly symmetrical figures as regular and
semiregular polyhedra was natural.
Below (Fig. 3.26) we examine the different representations of the dodeca
hedron used by Leonardo da Vinci in Pacioli’s Divina Proportione. Leonardo
used two methods, a method of Rigid Edges (Fig. 3.26a) and a method of Con
tinuous Faces (Fig. 3.26b). Comparison of these methods with the example of
the dodecahedron convincingly shows the advantage of the method of rigid edges.
Chapter 3
Regular Polyhedrons
173
The essence of the method of rigid edges consists in the fact that the faces of the
polyhedron are represented as “empty,” not continuous. Strictly speaking, the
faces are not represented at all; they exist only in our imagination. However, the
edges of the polyhedron are represented not by geometrical lines (which have
neither width nor thickness), but by the rigid threedimensional segments. Such
techniques of polyhedron repre
sentation allow the spectator to
accurately determine, first of all,
which edges belong to the front
and which belong to the back fac
es of the polyhedron (which is
practically impossible when the
edges are represented by geomet
rical lines). On the other hand, we
can look through a geometrical
a)
b)
body to see the body in perspec
Figure 3.26. Leonardo’s representations of the
tive and depth. It is clear that this
dodecahedron by the methods of (а) rigid edges
is impossible if we use the method
and (b) continuous faces in Pacioli’s Divina
Proportione
of continuous faces.
Below (Fig. 3.27) we see Leonardo’s representation of the truncated icosa
hedron by the method of rigid edges. At the top of the picture we see the Latin
inscription Ycocedron Abscisus (truncated icosahedron) Vacuus. The Latin word
Vacuus meant, that the faces of the polyhedron are represented as “empty,”
not continuous.
It is necessary to note that the representation of
polyhedra by the method of rigid edges began to be used
widely in science and works of art following Leonardo.
As an example, Johannes Kepler used the method of rigid
edges for the representation of the polyhedra from which
he constructs his Cosmic Cup (Fig. 3.11).
3.8.2. Pacioli’s Polyhedron
As we saw previously, Pacioli was one of the great
est mathematicians of 15th century Europe. He also
invented the principle of the socalled double record
Figure 3.27. Leonardo’s
representation of the
used now in all modern systems of bookkeeping. That
truncated icosahedron by
is why, he can be called “the father of modern book the method of rigid edges in
keeping.” However, the creative works of Luca Pacio Pacioli’s Divina Proportione
Alexey Stakhov
THE MATHEMATICS OF HARMONY
174
li, who was a very mysterious person in his time, have up to now been the source
of fierce disputes amongst historians of science. It is known that Luca Pacioli
was born in 1445 in the Italian city Borgo San Sepulcro. In his childhood he
was a pupil of the artist and mathematician Piero della Francesca. Later he was
a student at the University of Bologna, which during the 15th and 16th centu
ries was one of the best educational institutions in Europe (at various periods
Copernicus and Durer were its students). In 1472, Pacioli came back to his
native city Borgo San Sepulcro and started to write his best known book Sum
ma de Arithmetica, Geometria, Proportioni et Proportionalita. This book was not
published in Venice until 1494. In 1496, he was invited to Milan to give the
lectures on geometry and mathematics. Here is when he met Leonardo da Vin
ci. After reading Pacioli’s Summa, Leonardo stopped writing his own book on
geometry and started to illustrate Pacioli’s new book Divina Proportione.
Some researchers accuse the author of Div
ina Proportione of plagiarism of the unpublished
manuscript belonging to Pacioli’s teacher Pie
ro Della Francesca. However, it is rather doubt
ful, if we take into consideration that in the pe
riod when Pacioli was writing this book, he was
already a wellknown mathematician, and his
glory reverberated throughout all of Italy. Pa
cioli is well known to us today owing to the
Figure 3.28. Jacopo de Barbari’s
portrait (Fig. 3.28) painted of him by the Ital
painting of Luca Pacioli
ian artist Jacopo de Barbari (1440 1515). Bar
bari’s picture is revealing in several respects, first of all, in respect of the presen
tation of Luca Pacioli’s personality. Numerous compositional details in Barbari’s
picture are full of deep scientific sense. The artist demonstrates an understand
ing of the interrelation between Art and Science which was peculiar to Renais
sance experts. Pacioli in the robe of the Franciscan
monk is represented by standing behind the table of
geometrical tools and books (in the lower righthand
corner of the picture we can see a model of the
dodecahedron). The attention of Pacioli and the
handsome young man, who stands on the right and
somewhat behind Pacioli, is directed to the glass
model of the polyhedron that hangs in the upper left
hand corner of the picture. The choice of the poly
hedron is not accidental: it is Pacioli’s rhombic cube
Figure 3.29. Pacioli’s
octahedron (Fig. 3.29).
rhombic cubeoctahedron
Chapter 3
Regular Polyhedrons
175
3.8.3. Albrecht Durer
The identity of the young man, who is near to Pacioli in Jacopo de Bar
bari’s picture (Fig. 3.28), has often caused disputes amongst historians of art.
Some of them assert that this is Barbari’s selfportrait, others identifying the
individual with Albrecht Durer (1471 1528), German painter, draughtsman
and art theorist who is generally known as the greatest German Renaissance
artist. However, and this is very important in our context, Durer was amazed
by the art method of Barbari, who later created his pictures on the basis of a
deep study of the system of proportions, that is, strict use of particular ratios
of parts of the represented objects among themselves.
Durer was one of the first artists to start studying the laws of perspec
tive. He dreamed of meeting with the glorified Italian artists to study their
art works and to compete with them. With this purpose in mind Durer trav
eled to Italy in 1505. We do not know his teachers at the school of perspec
tive, but we do know that throughout his life Durer continued to teach at
this school. He wrote several books on the theory of arts. One of them, The
Painter’s Manual, was dedicated to geometry and perspective and was pub
lished in Nuremberg in 1525. His last work on human proportion was pub
lished in 1528 following his death.
Durer’s books are a serious scientific contribution to the theory of per
spective and stereometry of polyhedra. He described several Archimedean
Solids unknown at that time, and also developed and published for the first
time books that included spatial models of the planar unfolding of various
polyhedra, including the unfolding of the truncated icosahedron. Today a
similar unfolding of volumetric models of polyhedra is widely used to study
the elementary forms of crystals, the structures
of molecules (fullerenes, for example), and vi
ruses and so on.
In 1512 in the rough draft of his first trea
tise about proportions, Durer wrote: “All needs
of the person are satiated by the fleeting things
in the case of their surplus, so that they cause in
him disgust, except for a thirst for knowledge....
A desire to know a lot and through this to grasp
the essence of all things is inherent within us
from nature.” These words became a prologue
to Durer’s theoretical works. Art of that epoch Figure 3.30. Albrecht Durer’s
is often penetrated by a thirst for knowledge and
Melancholia
Alexey Stakhov
THE MATHEMATICS OF HARMONY
176
many representatives of that epoch became scientists and researchers. The
idea of the unity of artistic inspiration and mathematical theory is also re
flected in Durer’s wellknown picture Melancholia created in 1514. This pic
ture embodies the image of a person, who is near to God. The individual is
surrounded by various geometric tools (Fig. 3.30). The presence in the en
graving of a polyhedron (most likely, truncated rhombohedra), is certainly
not accidental.
3.8.4. Piero Della Francesca
We should start a list of the greatest experts of the Renaissance, who made
a concerted study of the geometry of polyhedra beginning with Piero Della
Francesca (1420 1492). We know a little about the life of Piero Della Frances
ca, who was a great Italian artist, art theorist and mathematician. It is known,
that he was born into the family of a craftsman in the small city of Borgo San
Sepulcro. He studied in Florence, then worked in some Italian cities, includ
ing Rome. Francesca’s creativity went far beyond the local art schools and
helped define the whole of art of the
Italian Renaissance. Fortunately, at
the beginning of the 20th century
the originals of three mathematical
manuscripts by Piero Della Frances
ca were discovered. They are now
housed in the Vatican Library. After
five centuries of obscurity, the glory
of this great mathematician has been
returned to Piero Della Francesca.
Now it is known for certain that
Francesca was the Renaissance’s
first expert, who rediscovered and
described in detail all Archimedean
Solids, in particular, the five trun
cated Platonic Solids, the truncat
ed tetrahedron, hexahedron, octa
hedron, dodecahedron, and, what is
especially important here, the trun
cated icosahedron.
Figure 3.31. Piero Della Francesca’s
Piero Della Francesca was both
The Baptism of Christ and its harmonious
a talented mathematician and great
analysis based on the golden section
Chapter 3
Regular Polyhedrons
177
artist. His art works express majestic solemnity, nobleness and harmony of
images, reasonableness of proportions, and clarity of perspective in their con
struction. His painting of The Baptism of Christ is one of the highlights of his
artistic style. Christ is placed in the center of the picture (Fig. 3.31). His feet
are being washed by the river water; his hands are combined in a gesture of
Catholic prayer. Near to Christ we see John the Baptist, who pours water
from a dish on Christ’s head. The dove, representing the Holy Spirit, descends
from above the head of Christ.
3.8.5. Art of Intarsia
At the end of the 15th and beginning of the 16th centuries the art of Intar
sia, a special kind of inlay, a mosaic, constructed from thousands of fine slices of
various species of trees, was very pop
ular in Northern Italy. The mosaic
created by Fra Giovanni da Verona
(1457 1525) for the church Santa
Maria in Verona in roughly 1520, is
an example of intarsia (Fig. 3.32).
The image of halfopened shutters
creates on the planar mosaic a spa
tial effect, which increases through
the representation of various poly
hedra (including the truncated
icosahedron) by using Leonardo’s Figure 3.32. Fra Giovanni da Verona’s intarsia
technique of rigid edges.
created for the church Santa Maria in Verona
3.8.6. Salvador Dali’s Last Supper
Now, let us consider an example of the representation of polyhedra by the
wellknown 20th century artist Salvador Dali (1904 1989). This great Spanish
artist is one of the best known representatives of surrealism. An excellent art
ist, Dali created images similar to dreadful visions, called by him “drawing
pictures of dreams.” Some of the most common repeat images, for example
hours, which lose their forms under the sun beams, became Dali’s logo. Dali’s
creativity continues to cause disputes (e.g. some critics even suggest that af
ter 1930 he did not create anything of real worth). However, in 1955 Dali
created one of his best known pictures, Last Supper (Fig. 3.33).
This big picture is an original masterpiece of painting. Geometrical ra
tionalism testifies to the invincible belief in the sacred force of number. In
Alexey Stakhov
THE MATHEMATICS OF HARMONY
178
the center of the big horizontal
picture (167×288) we see Christ.
Together with his pupils he sits
at the table as God’s son descend
ed from the Heavens (Dodecahe
dron) upon the Earth. The apos
tles are represented with heads
held low inclined to the table.
They appeared as if in deep wor
ship of Christ. The dodecahe
Figure 3.33. Salvador Dali’s Last Supper
dron in this picture plays a cen
tral role because it personifies the spiritual harmony, moral cleanliness and
greatness of character.
3.8.7. Escher’s Creative Work
Maurits Cornelis Escher (1898 1972) is one of the world’s most famous
graphic artists. His art continues to amaze millions of people all over the world.
In his works we recognize his keen observation of the world around us and the
expressions of his own fantasies. Escher shows us that reality is wonderful and
fascinating. He is most famous for his socalled impossible structures, such as
Ascending and Descending, Relativity, and his Transformation Prints such as
Metamorphosis I, Metamorphosis II and Metamorphosis III, Sky & Water and
Reptiles (see www.mcescher.com).
Escher’s creative work was highly esteemed by many scientists, in partic
ular, by mathematicians and crystallographers. At the International Crystal
lographic Congress in Cambridge (1960) the exhibition of Escher’s pictures
became a sensation attracting the special attention of the crystallographers.
What do art and crystallography hold in common? The question arises as to
whether Maurits Cornelis Escher intuitively discovered the laws of symme
try, those laws which dominate over crystals and define their external shape,
nuclear structure and physical properties, and then illustrated these laws in
his pictures. Escher took great interest in periodic figures and drew up the
mosaic patterns of the repeating figures in his pictures. He inserted one image
in another so that identical figures are periodically repeated without leaving
any empty space between them. In fact, it is based on the same law, according
to which the particles in crystal structures are placed, namely, the law of the
densest packing: the periodic recurrence of identical groups of particles with
out intervals and infringements.
Chapter 3
Regular Polyhedrons
179
3.9. Application of the Golden Mean in Contemporary Art
3.9.1 Abstract Art by Astrid Fitzgerald
Astrid Fitzgerald is an internationally acclaimed artist, born and educated
in Switzerland and now residing and working in New York. Her works are rep
resented in major public museums and private collections in the United States,
Europe and Asia. Fitzgerald’s installation, Amish Quilts, was chosen by the judges
to represent the United States at the Artcanal Exposition (2002) in Switzerland.
She is also the author of An Artist’s Book of Inspiration – A Collection of Thoughts
on Art, Artists and Creativity (1996) and Being Consciousness Bliss – A Seeker’s
Guide (2002), both published by Lindisfarne Books. Fitzgerald refers to her re
cent work as Cosmic Measures – a phrase that expresses
her continuing search for the true nature of things. Here
she began to work with the fundamental laws of geo
metric forms that include the Golden Mean – the uni
versal principle that underlies Nature from the spiral of
our DNA to that of our galaxy. These harmonious pro
portions have fascinated philosophers, architects and
artists for millennia. Fitzgerald embodies the golden
mean within her abstract pictures.
In Figs. 3.34 we can see some pictures of Astrid
Astrid Fitzgerald
Fitzgerald based on the Golden Mean.
Figure 3.34. Abstract art of Astrid Fitzgerald
Alexey Stakhov
THE MATHEMATICS OF HARMONY
180
3.9.2. The World of Matjuska Teja Krasek
Matjuska Teja Krasek obtained her B.A. degree in painting from the Arthose–
College for visual arts in Ljubljana, Slovenia. Her theoretical and practical work
focuses primarily on symmetry as a connecting concept between art and science.
Through her art, Krasek wishes to convey her experience and feelings that are
connected with her research results from various disciplines. She wants to convey
this to all who are interested in exploring the way and functioning of our universe
and nature. In this manner she wishes to contribute to the awareness of certain
characteristics such as the various kinds of symmetry, the
golden mean, and the Fibonacci sequence’s connection to
nature, natural science and art. She also explores in her works
how the use of various formal elements of artistic expression
(lines, colors, structures, etc.) can influence the stability of
art work. She uses contemporary computer technology as
well as a classical painting technique. And her artworks have
been represented at many international exhibitions and pub
lished in international journals (Leonardo Journal, Leonardo
online and so on).
Matjuska Teja Krasek
In Fig. 3.35 we can see some pictures of Matjuska Teja Krasek.
Figure 3.35. The art works of Matjuska Teja Krasek
3.9.3. The Geometric Art of John Michell
John Michell, born in 1933, was educated at Eton and Cambridge and
published his first book in 1967. The author of 12 books, he is a specialist in
Chapter 3
Regular Polyhedrons
181
sacred geometry and the geometry of reconciliation. With
The View over Atlantis (1969) and City of Revelation (1972)
Michell helped to change the worldviews of a whole gen
eration by illuminating the science, culture and wisdom of
past civilizations. Michell’s City of Revelation was sum
marized and extended with further research in The Dimen
sions of Paradise, in which the proportions and symbolic
John Michell
numbers of ancient cosmology were explained.
Michell’s books, Twelve Tribe Nations and Science of Enchanting the Land
scape (1991), the latter with Christine Rhone, explain that throughout the
history of civilization an ideal social order harmonically related to nature and
the zodiac was imposed upon the landscapes of the world. This is depicted, for
example, by the 12 tribe divisions of people and the careful alignment of var
ious holy places.
Fig. 3.36. The Geometric Art of John Michell
3.9.4. Quantum Connections by Marion Drennen
Marion Drennen is a Louisiana Artist, who received her Bachelor in Fine
Arts from LSU, working in Acrylic on Board. These paintings incorporate the
concepts of Number and Quantum Physics in an effort to evoke a sense of con
nectedness across time and space, within ourselves and in our relationships.
In her artistic statement she describes her artistic
process – “Being an idea person, an avid reader and re
searcher, my paintings manifest from contemplations on
a variety of subjects – Mathematics, The Golden Ratio,
Quantum Physics, and Spiritual. I glimpse the connec
tions and begin to draw thumbnail sketches around the
edges of my notes. The internal dialogue is about the con
Marion Drennen
cept, then words gradually disappear and visual elements
Alexey Stakhov
THE MATHEMATICS OF HARMONY
182
expand until only sketches are coming off the end of the pencil. Later, within
the painting, words may reappear.” She compares her creation process with a
dance – “I used to dance. There’s something exquisite about losing yourself in
movement. Sometimes, when I’m painting, I’ll start to dance. I almost always
paint to music. After working with the Golden Ratio for several years, I now
have custom painting surfaces made that allow me to paint within that for
mat. It sets the stage. It is the architecture, the structure on which I begin to
build. I break up the space and then insert my one or two shapes, the initial
idea, and then the dance begins.”
In Fig. 3.37 we represent some pictures of Marion Drennen from her Quan
tum Connections exhibit that was showing at the Brunner Gallery in the Shaw
Center for the Arts.
Fig. 3.37. Quantum Connections by Marion Drennen.
3.10. Conclusion
The regular and semiregular polyhedra have been known from antiquity.
The regular polyhedra got the name Platonic Solids, because they played such
an important role in Plato’s cosmology. According to Plato, the atoms of the
Universe’s “Basic Elements” have the form of the Platonic Solids (Fire – a Tet
rahedron, Earth – a Hexahedron or Cube, Air – an Octahedron, Water –an
Icosahedron). The Dodecahedron was considered to be the primary figure of
the Universe, expressing the Universal Intellect and the Harmony of the Uni
verse. The semiregular polyhedra are named Archimedean Solids. A geometric
theory of the Platonic Solids was presented in the 13th or final Book of Euclid’s
Chapter 3
Regular Polyhedrons
183
Elements. This is the reason why the ancient Greek mathematician Proclus, a
commentator on Euclid, put forward the hypothesis about the true purpose of
Euclid writing The Elements. In Proclus’ opinion, Euclid wrote his Elements to
give a full and systematized theory of geometric construction of the “ideal” geo
metric figures, in particular, the five Platonic Solids. Thus, we can consider The
Elements of Euclid to be the first historical geometrical theory of the Harmony
of the Universe, based upon the golden section (division in the extreme and
mean ratio) and the Platonic Solids! Since antiquity the Platonic and
Archimedean Solids have been a source of many scientific hypotheses, theories
and discoveries. The surprising coincidence of the numerical characteristics of
the dodecahedron (12 faces, 30 edges and 60 planar angles on its surface) with
the main cycles of the Solar System (12year cycle of Jupiter, 30year cycle of
Saturn, and 60year basic cycle of the Solar System) apparently became one
reason why the numbers 12, 30, 60 and 360=12×30 were used by the Ancient
Egyptians in their calendar and their systems of time and angle measurement.
In the 19th century the prominent mathematician Felix Klein began to consid
er the regular Icosahedron as the main geometric object, from which the branches
of the five mathematical theories follow, namely, geometry, Galois’ theory, group
theory, theory of invariants and differential equations. Shechtman’s quasicrys
tals were based on the icosahedron and fullerenes (1996 Nobel Prize in chemis
try). These are brilliant confirmations of the role of the Platonic and
Archimedean Solids, and therefore the golden section, in modern physics.
This page intentionally left blank
Part II
Part II.
Mathematics of Harmony
Alexey Stakhov
THE MATHEMATICS OF HARMONY
186
Chapter 4
Generalizations of Fibonacci Numbers and the
Golden Mean
4.1. A Combinatorial Approach to the Harmony of Mathematics
4.1.1. Mathematical, Aesthetic and Artistic Understanding of Harmony
In the Introduction we mentioned that the Harmony Problem is one of the
“key” problems of mathematics entering the scene at its very origin. But what
does a concept of “harmony” mean?
The Russian philosopher Shestakov, one of the best researchers in the field,
pointed out three basic understandings of “harmony” that have been devel
oped in science and aesthetics since antiquity [7]:
1. Mathematical Understanding of Harmony or Mathematical
Harmony. In this sense, harmony is understood as equality or proportion
ality of parts one to another and the parts to the whole.
2. Aesthetic Harmony. In contrast to the mathematical harmony, the
aesthetic harmony is not quantitative, but qualitative notion and express
es the internal nature of things. The aesthetic harmony is connected with
aesthetic excitements and estimations. Most precisely this type of harmo
ny is shown at perception of beauty of Nature.
3. Artistic Harmony. This type of harmony is connected with art. Artistic
harmony is an actualization of the harmony principle in the realm of art.
In the present book, our attention is concentrated on Mathematical
Harmony. It is clear that mathematical harmony is expressed in the form of
certain numerical proportions. Shestakov emphasizes that mathematical har
mony “attracts attention to its quantitative side and is indifferent to qualita
tive originality of the parts forming conformity.... The mathematical under
standing of the harmony fixes, first of all, quantitative definiteness of the har
mony, but it does not express aesthetic quality of the harmony, its expressive
connection with beauty.”
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
187
Thus, in this book we consciously restrict the area of our research to the
mathematical understanding of harmony. This approach is fruitful in mathe
matics and allows us to create a Mathematical Theory of Harmony or Mathe
matics of Harmony that extends the area of the mathematical models of the
harmonic processes of Nature.
4.1.2. Concept of the Mathematical Theory of Harmony
We can ask a question: how does one create a Mathematical Theory of
Harmony? As is known, mathematics studies the quantitative aspect of this
or that phenomenon. Starting with a mathematical analysis of the concept of
harmony, we should concentrate our attention on the quantitative aspects of
harmony. What is the quantitative aspect of this concept?
There are many definitions of the harmony concept. However, the major
ity of them are reduced to the following definition that is given in The Great
Soviet Encyclopedia:
“Harmony is proportionality of parts and the whole, a combination of the
various components of the object in the uniform organic whole. It is the inter
nal order and measure obtained in the harmony external expression [of the
harmony].”
In the article Harmony in Nature and Art [149], the Russian crystallogra
pher Shubnikov compared harmony with the Order studied by science to dis
cover Nature’s laws. He writes: “Law, Harmony, Order underlie not only sci
entific work but also any work of art.”
Now let us analyze a question of the origin and meaning of the word “harmo
ny.” “Harmony” has a Greek origin. The Greek word αρµουια has the following
meaning: connection, consent. The analysis of the word of “harmony” and its defi
nitions demonstrate that the most important, key notions, which underlie this
concept, are the following: Connection, Consent, Combination, Order.
We can ask a question: what branch of mathematics studies these concepts?
A search for the answer leads us to Combinatorial Analysis. “Combinatorial anal
ysis studies the various kinds of combinations and connections, which can be
formed from the elements of some finite set. The term ‘combinatorial’ is derived
from the Latin word combinary which means to combine or connect.” [150]
It follows from this consideration that both the Latin word combinary and
the Greek word αρµουια, in essence, have the same meaning, namely, Combi
nation and Connection. It allows us to put forward the following hypothesis:
the “Laws of combinatorial analysis” can be used for the analysis of the Har
mony concept from the quantitative point of view.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
188
Therefore, the main concepts underlying the Harmony Mathematics and
the combinatorial analysis, in essence, coincide. And this allows us to use com
binatorial analysis as the main mathematical theory for the analysis of the
harmony concept from the quantitative point of view. This is the main idea
behind the research, in which the author made an attempt to create the Fun
damentals of the Harmony Mathematics based upon combinatorial analysis.
Thus, Harmony Mathematics is a mathematical theory studying the no
tion of harmony from a quantitative point of view. Its main goal is to study
mathematical laws, mathematical proportions, which underlie the Harmony
of the Universe.
4.1.3. An Analogy between the Theory of Information and Mathematics
of Harmony
Are there similar theories in modern science? The Mathematical Theory of
Information developed by Claude Shannon [151] is possibly a brilliant exam
ple of a similar theory. Information is a complex and interdisciplinary concept
similar to Harmony. Both information and harmony are nonmaterial and om
nipresent. In spite of its nonmaterial character, the Mathematical Theory of
Harmony may be considered to be an original mathematical theory similar to
the Theory of Information.
By developing a Theory of Information, Claude Shannon used the concept
of Probability as the starting point of the theory. The concept of Entropy, based
on the notion of probability, became the underlying basic concept of the The
ory of Information. It is necessary to emphasize that Shannon’s Theory of In
formation is a mathematical theory and can be effectively used, first of all, for
the quantitative analysis of any informational system. In his wellknown arti
cle Bandwagon [151] Shannon warned that one must be cautious applying
this theory to other areas of human activity.
Shannon’s Theory of Information is sometimes considered to be a branch
of probability theory. By continuing the analogy between Shannon’s Theory
of Information [151] and Harmony Mathematics, it is possible to consider the
Harmony Mathematics as a special branch of combinatorial analysis.
From this point of view, it becomes imperative to answer the question
about practical application of the Harmony Mathematics. What are the areas
of the effective application of this theory? By answering this question, it is
important to emphasize that the most effective areas are those where the quan
titative aspects of the Harmony are most important, such as theoretical phys
ics, computer science, biology, botany, economics, and so on.
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
189
4.2. Binomial Coefficients and Pascal Triangle
4.2.1. The Main Concepts of Combinatorial Analysis
Combinatorial analysis studies different kinds of combinations that can
be built up from the elements of some set. The term “combinatorial analysis”
originates from the Latin word combinare that has the meaning to combine
or connect.
Some elements of combinatorial analysis were known in India already in
the 2nd century AD. The Indian mathematicians knew how to calculate num
bers Cnm called Combinations from n elements by m, and they knew the fol
lowing formula:
(4.1)
Cn0 + Cn1 + ... + Cnn = 2n .
A theory of binary codes, the basis of modern computers, is an example of
the effective application of the formula (4.1). We can consider a set of the
ndigit binary words starting from the code combination 00 … 0 and ending by
the code combination 11 … 1. As is known, the number of elements of this set
is equal to 2n. We can divide this set into the (n+1) disjoint subsets. Then we
can refer all binary words consisting only of 0’s to the first subset. It is clear
that the only code combination 00 … 0 satisfies this condition, that is, the
number of the elements of this subset is equal to Cn0 = 1. Then, we can refer to
the second subset all the binary words containing only 1 and the (n1) 0’s. It
is clear that the number of elements of this subset is equal to Cn1 . We can refer
to the (m+1)th subset all the ndigit binary words containing m 1’s and (n
m) 0’s; the number of code combinations of this subset is equal to Cnm . At last,
we can refer to the (n+1)th subset all the ndigit binary words which contain
only 1’s. It is clear that the only code combination 11 … 1 satisfies this condi
tion, that is, the number of elements of this subset is equal to Cnn = 1. From this
reasoning the validity of the formula (4.1) follows.
The term “combinatorial analysis” began to be used after publication
in 1666 of Leibniz’s Reasoning about Combinatorial Art; in this book he
gave for the first time scientific substantiation of the theory of combina
tions and permutations. Bernoulli introduced for the first time a notion
of Distribution in the second part of his famous book Art of Guessing pub
lished in 1713. He introduced and used in our sense the term Permuta
tion. The term Combination was introduced by Pascal in his Treatise about
the Arithmetical Triangle (1665).
Alexey Stakhov
THE MATHEMATICS OF HARMONY
190
Thus we have the wellknown formula
Cnm =
n!
m !(n − m)! ,
(4.2)
where n!=1×2×3×…×n is a Factorial of n.
4.2.2. Binomial Formula
Now, write the following formulas:
(a + b) = 1
1
( a + b ) = 1a + 1b
( a + b )2 = 1a2 + 2ab + 1b2
3
( a + b ) = 1a3 + 3a2b + 3ab2 + 1b3 .
0
(4.3)
Note that the first two formulas from (4.3) are trivial; the other two for
mulas are wellknown from secondary school.
We can ask the question: how do we calculate the binom (a+b)n? The
wellknown mathematical formula called the Binomial Formula gives the an
swer to this question:
( a + b ) = a n + Cn1 a n −1b + Cn2 a n −2b2 + ... + Cnk a n − k b k + ... + Cnn −1abn −1 + bn .
n
Cnk
(4.4)
are named Binomial Coefficients or Binomial
Here the numbers
Factors.
Sometimes the discovery of formula (4.4) was attributed to Newton. How
ever, long before Newton, the mathematicians of many countries, in particu
lar, the Arab mathematician Al Kashi, the Italian mathematician Tartalja, the
French mathematicians Fermat and Pascal, knew this formula. Newton’s mer
it consists in the fact that he derived this formula for the case of any real num
ber n, that is, he proved that the formula (4.4) is true when n is rational or
irrational, positive or negative.
The formulas (4.3) are special cases of a general formula (4.4). In particu
lar, for the case n=1 the formula (4.4) is reduced to the following:
( a + b )1 = C10 a + C11b = 1a + 1b,
whence it appears that C10 = 1 and C11 = 1.
For the case n=2, the formula (4.4) takes the following form:
( a + b ) = C20 a2 + C21ab + C22 = 1a2 + 2ab + 1b2 ,
2
whence it appears that C20 = 1, C21 = 2, C22 = 1.
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
191
Thus, we can get the decomposition (4.4) which can be very easy if we
know how to calculate the binomial coefficients Cnk .
4.2.3. Pascal Triangle
The French 17th century mathematician Blaise Pascal (16231662) sug
gested an original method for the calculation of the coefficients Cnk for any
arbitrary nonnegative integers n and k.
In addition to the properties (4.1) and (4.2), the binomial coefficients Cnk
have a number of remarkable properties that are given here without a proof:
Cn0 = Cnn = 1
(4.5)
Cnk = Cnn − k
(4.6)
(4.7)
Cnk+1 = Cnn −1 + Cnk .
The last property (4.7) is also named the Pascal Law.
Using the recursive relation (4.7), Pascal had offered an orig
inal method for the calculation of binomial factors that are
based on their disposition in the form of a special numerical
table called Pascal’s Triangle.
Let us examine an infinite table of numbers constructed
according to the Pascal Law (4.7).
Blaise Pascal
(16231662)
The top of the indicated table (Fig. 4.1) that is named
Zerorow, consists of the only binomial coefficient C00 = 1. The next row – the
1st row consists of two binomial coefficients C10 = C11 = 1. Each succeeding
row can be constructed from the preceding row according to the rules (4.5)
(4.7). It is easy to prove the following properties of Pascal triangle:
1. The sum of the binomial coefficients of the nth row of Pascal triangle
is equal to 2n what corresponds to the identity (4.1).
2. All rows of Pascal triangle are symmetric relative to the binomial coef
ficient C00 = 1 of the zerorow that corresponds to the property (4.6).
The above Pascal Trian
gle appeared for the first time
1
1
1
in Pascal’s Treatise about Ar
1
2
1
ithmetical Triangles written in
1
3
3
1
1
4
6
4
1
1665. However, one century
1
5
10
10
5
1
prior to the publication of
1
6
15
20
15
6
1
1
7
21
35
35
21
7
1
Pascal’s Treatise, this numer
1 8
28
56
70
56
28
8 1
1 9 36
84
126
126
84
36 9 1 ical table (but in rectangular
Figure 4.1. Pascal Triangle
form rather than triangular)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
192
was described in the General Treatise about Number and Measure written by
the Italian mathematician Nikola Tartalja (15001557). Tartalja’s Treatise was
published immediately after his death.
Here the first or top row
Table 4.1. Tartalja’s Rectangle
and the first column on the
1
1
1
1
1
1
left consist only of 1’s; each
1
2
3
4
5
6
“internal” number of other
1
3
6
10
15
21
1
4
10
20
35
56
rows is equal to the sum of
1
5
15
35
70
126
two numbers, the first one
1
6
21
56
126
252
that stands in the same row
on the left to it and the second one that stands in the preceding row above it.
This table of binomial factors is called Tartalja’s Rectangle (Table 4.1).
4.3. The Generalized Fibonacci pNumbers
4.3.1. Rectangular Pascal Triangle
There are many forms of the representation of the Pascal Triangle, for exam
ple, in the form of an isosceles triangle, in the form of a rectangular table (Tartal
ja’s Rectangle), etc. We will examine the socalled Rectangular Pascal Triangle
that can be represented by the following table of binomial coefficients (Fig. 4.2).
The rows of the Pascal Triangle are numbered C 0 C 0 C 0 C 0 C 0 C 0 … C 0
n
0
1
2
3
4
5
1
1
1
1
1
1
from top to bottom. The binomial coefficients
C1 C 2 C 3 C 4 C 5 … C n
2
2
2
2
2
C00 = C10 = C20 = ... = Cn0 = 1 make up a “zero” row.
C2 C3 C4 C5 … C n
3
3
3
3
Every nth row starts with the binomial coefficient
C3 C4 C5 … C n
4
4
4
of the kind Cnn = 1 ( n = 0, 1, 2, 3, ...) .
C4 C5 … C n
5
5
The columns of the Pascal Triangle are num
C5 … C n
Figure 4.2.
bered from left to right; the first lefthand column Rectangular Pascal % #
n
Cn
Triangle
that consists of the only binomial coefficient
C00 = 1 is called the Zerocolumn. The nth col
umn (n=0,1,2,3,…) includes the following binomial coefficients:
(
)
Cn0 , Cn1 , Cn2 , ..., Cnk , ..., Cnn − k , ..., Cnn ,
where Cnk = Cnn − k .
As mentioned above, Pascal Triangle is based on the recursive relation (4.7).
What is the correlation between Pascal Triangle and Fibonacci numbers?
In the second half of the 20th century many Great mathematicians (Martin
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
193
Gardner [12], George Polya [17], Alred Renyi [23] and others) independently
one after another discovered the connection of Fibonacci numbers with Pascal’s
Triangle and binomial coefficients. This discovery demonstrates a fundamental
connection of the Harmony Mathematics based on Fibonacci numbers and the
golden mean with combinatorial analysis and outlines a way for the future gen
eralization of Fibonacci numbers. Below we will demonstrate a surprisingly sim
ple mathematical regularity that connects Pascal Triangle and Fibonacci num
bers. A generalization of this regularity resulted in the mathematical discovery
called the Generalized Fibonacci pNumbers [19, 20].
4.3.2. Pascal pTriangles and Fibonacci pNumbers
Let us examine the Rectangular Pascal Triangle represented in numerical
form (Fig. 4.3).
We can name the given table of binomial co
1 1 1 1 1 1 1 1 1 1
efficients Pascal 0Triangle (the meaning of this
1 2 3 4 5 6 7 8 9
1 3 6 10 15 21 28 36
definition will become clear below). If we sum the
1 4 10 20 35 56 84
binomial coefficients of the Pascal 0Triangle by
1 5 15 35 70 126
columns starting from the 0column,
1 6 21
1 56 126
1 7 28 84
then according to (4.1) we obtain the binary se
1 8 36
quence:
1
9
1
1 2 4 8 16 32 64 128 256 512
(4.8)
1, 2, 4, 8, 16, ..., 2n , ....
Now, we do some “manipulations” around the
Figure 4.3. Pascal 0Triangle
Pascal 0Triangle. We move each row of the Pas
cal 0Triangle one column to the right with respect to the previous row. As a result
of such move, we obtain a table called Pascal 1Triangle (Fig. 4.4).
Now, sum the binomial coefficients of the Pascal 1Triangle in each column.
To our amazement, we find that this summation results in the Fibonacci numbers:
1,1,2,3,5,8,13,21,…,Fn+1,….
(4.9)
where Fn+1 is the (n+1)th Fibonacci number given by the following recursive
relation:
(4.10)
Fn+1= Fn+Fn1.
The numerical sequence (4.9) is generated by the recursive relation (4.10)
at the seeds:
F1=F2=1.
(4.11)
If we move in the initial Pascal 0Triangle (Fig. 4.1) the binomial coeffi
cients of each row by p columns to the right with respect to the previous row
Alexey Stakhov
THE MATHEMATICS OF HARMONY
194
(p=0,1,2,3,…), we get the numerical table named 1 1 1 1 1 1 1 1 1 1 1 1
1 2 3 4 5 6 7 8 9 10
Pascal pTriangle. It is clear that the Pascal 0
1 3 6 10 15 21 28 36
Triangle that corresponds to the case p=0 is the
1 4 10 20 35 56
initial Pascal Triangle (Fig. 4.3). The Pascal 1
1 5 15 35
1 6
Triangle is represented in Fig. 4.4. The Pascal
1
1
2
3
5
8
13
21
34
55
89
144
pTriangles for the cases p=2 and p=3 have the
forms shown in Fig. 4.5.
Figure 4.4. Pascal 1Triangle
Now, sum the binomial coefficients in each column of the Pascal 2 and 3
Triangles. As a result, we obtain two new numerical sequences:
1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, ...
(4.12)
1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 26, ...
(4.13)
Pascal 2Triangle
Pascal 3Triangle
1 1 1 1 1 1 1 1 1 1 1 1 1
1 2 3 4 5 6 7 8 9 10
1 3 6 10 15 21 28
1 4 10 20
1
1 1 1 2 3 4 6 9 13 19 28 41 60
1 1 1 1 1 1 1 1 1 1 1 1 1
1 2 3 4 5 6 7 8 9
1 3 6 10 15
1
1 1 1 1 2 3 4 5 7 10 14 19 26
Figure 4.5. Pascal 2Triangle and 3Tirangle
Denote by F2(n) and F3(n) the nth elements of the sequences (4.12) and (4.13),
respectively. It is easy to see the following regularities in the numerical sequences
(4.12) and (4.13) that can be expressed by the following recursive relations
F2(n)= F2(n1)+F2(n3) for n≥4
at the seeds
(4.14)
F2(1)=F2(2)=F2(3)=1
and by the recursive relation
(4.15)
F3(n)=F3(n1)+F3(n4) for n≥5
at the seeds
(4.16)
(4.17)
F3(1)=F3(2)=F3(3)=F3(4)=1.
Thus, as a result of this examination we have found two new numerical se
quences. The first of them that is given by the recursive relation (4.14) at the
seeds (4.15) is named the Fibonacci 2numbers and the second one that is given
by (4.16) at the seeds (4.17) is named the Fibonacci 3numbers.
In the general case, for arbitrary р if we sum the binomial coefficients of
each column of the Pascal pTriangle, we obtain the numerical sequences giv
en by the following recursive relation:
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
195
Fp(n)=Fp(n1)+Fp(np1) for n≥p+1
at the seeds
(4.18)
(4.19)
Fp(1)=Fp(2)=...=Fp(p+1)=1.
The numerical sequences that correspond to the recursive relation (4.18)
and (4.19) are named [20] the Fibonacci pnumbers.
4.3.3. Partial Cases of the Fibonacci pNumbers
It is clear that for the case p=0 the recursive relation (4.18) and the seeds
(4.19) takes the following form:
F0(n)=F0(n1)+F0(n1) for n ≥2
(4.20)
F0(1)=1.
(4.21)
It is easy to guess that the recursive relation (4.20) at the seed (4.21)
generates the binary sequence (4.8) that is a special case of the Fibonacci
pnumbers for p=0.
Let us examine the case p=1. For this case the recursive relation (4.18)
and the seeds (4.19) are reduced to the following:
F1(n)=F1(n1)+F1(n2) for n≥3
(4.22)
F1(1)=F1(2)=1.
(4.23)
Comparing these formulas with the recursive relation for the classical Fi
bonacci numbers (4.10) and (4.11), we can conclude that the Fibonacci 1
numbers coincide with the classical Fibonacci numbers, that is, F1(n)=Fn.
At last, we find that for the case p=∞ the sequence of the Fibonacci p
numbers consists only of the unities: {1,1,1,…}.
4.3.4. A Representation of the Fibonacci pNumbers by the Binomial
Coefficients
In the above we have examined the formula (4.1) that allows us to repre
sent binary numbers by the binomial coefficients. Analyzing the Pascal 1Tri
angle (Fig. 4.4), it is easy to derive the mathematical formula that allows us to
represent the Fibonacci 1numbers by the binomial coefficients:
F1 ( n + 1) = Cn0 + Cn1 −1 + Cn2−2 + Cn3−3 + Cn4−4 + ....
(4.24)
This means that there are two ways to calculate the Fibonacci numbers,
namely, by using the recursive relation (4.10) at the seeds (4.11) or by using
the formula (4.24).
Alexey Stakhov
THE MATHEMATICS OF HARMONY
196
For example, using the formula (4.24), we can represent the Fibonacci num
ber F1(7)=13 as follows:
(4.25)
F1 ( 7 ) = C60 + C51 + C42 + C33 + C24 + ....
4
Note that the binomial coefficient C2 in the sum (4.25) as well as all of the
binomial coefficients following C24 are equal to 0 identically. This means that
the expression (4.25) is the following finite sum of the binomial coefficients:
(4.26)
F1 ( 7 ) = C60 + C51 + C42 + C33 = 1 + 5 + 6 + 1 = 13.
Studying the Pascal pTriangle, we can represent the Fibonacci pnum
ber Fp(n+1) by the binomial coefficients as follows:
Fp ( n + 1) = Cn0 + Cn1 − p + Cn2−2 p + Cn3− 3 p + Cn4− 4 p + ....
(4.27)
Note that the known formula (4.1) is a partial case of (4.27) for p=0 and
the formula (4.24) is a partial case of (4.27) for p=1.
4.3.5. The “Extended” Fibonacci рNumbers
Until now we have studied the Fibonacci pnumbers Fp(n) given for the
positive values of n. In Chapter 2 we extended the classical Fibonacci num
bers into the side of the negative values of n. By analogy, we can find the ex
tended Fibonacci pnumbers, if we extend the Fibonacci pnumbers to the
side of the negative values of n. With this purpose, we will find some general
properties of such extended sequences. For the calculation of the Fibonacci p
numbers Fp(n) corresponding to the nonnegative values of n=0,1,2,3,… we
use the recursive relation (4.18) and the seeds (4.19). Let us represent the
Fibonacci pnumber Fp(p+1) in the form (4.18) as follows:
Fp(p+1)=Fp(p)+Fp(0).
(4.28)
According to (4.19), we have: Fp(p+1)=Fp(p)=1. This means that Fp(0)=0.
Continuing this process, that is, representing the Fibonacci pnumbers
Fp(p), Fp(p1),…, Fp(2) in the form (4.18), we have:
Fp(0)=Fp(1)=Fp(2)=...=Fp(p+1)=0.
(4.29)
Now, represent the Fibonacci pnumber Fp(1) in the form (4.18):
Fp(1)=Fp(0)+Fp(p).
As Fp(1)=1 and Fp(0)=0, we get from (4.30):
(4.30)
(4.31)
Fp(p)=1.
Representing the Fibonacci pnumbers Fp(0), Fp(1), …, Fp(p+1) in the
form of (4.18) , we get:
Fp(p1)=Fp(p2)=...=Fp(2p+1)=0.
(4.32)
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
197
Table 4.2. The “Extended” Fibonacci рNumbers
N
6 5 4 3 2 1 0 1 2 3 4 5 6 7
F1(n) 8 5 3 2 1 1 0 1 1 2 3 5 8 13
F2(n) 4 3 2 1 1 1 0 0 1 0 1 1 1 2
F3(n) 3 2 1 1 1 1 0 0 0 1 0 0 1 1
F4(n) 2 1 1 1 1 1 0 0 0 0 1 0 0 0
F5(n) 1 1 1 1 1 1 0 0 0 0 0 1 0 0
By continuing this process,
we can obtain all values of the
Fibonacci pnumbers Fp(n) for
the negative values of n. Table
4.2 gives some values of the “ex
tended” Fibonacci pnumbers
for the cases p=1, 2, 3, 4, 5.
4.3.6. Some Identities for the Sums of the Fibonacci рNumbers
Once again, let us examine the recursive relation (4.14). Decomposing
the Fibonacci 2number F2(n1) in (4.14) according to the same recursive re
lation (4.14), that is, representing F2(n1) in the form F2(n1)=F2(n2)+F2(n
4), we can represent the recursive relation (4.14) as follows:
F2(n)=F2(n2)+F2(n3)+F2(n4).
(4.33)
This means that the sum of the three successive Fibonacci 2numbers is
always equal to the Fibonacci 2number, which is two positions from the se
nior Fibonacci 2number of the sum.
Now, consider the recursive relation for the Fibonacci 3numbers given
by (4.16). Decomposing the Fibonacci 3number F3(n1) in (4.16) according
to the same recursive relation (4.16), we can represent the recursive relation
(4.16) as follows:
(4.34)
F3(n)=F3(n2)+F3(n4)+F3(n5).
Decomposing the number F3(n2) in (4.34) according to the recursive re
lation (4.16), we can represent (4.34) as follows:
F3(n)=F3(n3)+F3(n4)+F3(n5)+F3(n6),
(4.35)
that is, the sum of the four sequential Fibonacci 3numbers is always equal to
the Fibonacci 3number, which is three positions from the senior Fibonacci 3
number of the sum.
If we use a similar approach to the Fibonacci рnumbers in the general
case, we obtain the following general identity:
(4.36)
Fp(n)=Fp(np)+Fp(np1)+Fp(np2)+...+Fp(n2p).
Note that the identity (4.36) is valid for all “extended” Fibonacci pnum
bers Fp(n) when n takes the values from the set: {0,±1,±2,±3,…}.
Now, let us consider the sum of the first n Fibonacci pnumbers:
Fp(1)+Fp(2)+Fp(3)+...+Fp(n).
(4.37)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
198
To get the required result, we write the basic recursive relation (4.18) for
the Fibonacci pnumbers as follows:
Fp(n)=Fp(n+p+1)Fp(n+p).
By using (4.38), we can write the following equalities:
(4.38)
Fp (1) = Fp ( 2 + p ) − Fp (1 + p )
Fp ( 2 ) = Fp ( 3 + p ) − Fp ( 2 + p )
Fp ( 3 ) = Fp ( 4 + p ) − Fp ( 3 + p )
...
Fp ( n − 1) = Fp ( n + p ) − Fp ( n + p − 1)
Fp ( n ) = Fp ( n + p + 1) − Fp ( n + p ) .
Summing term by term the lefthand and righthand parts of these equal
ities and taking into consideration that Fp(1+p)=1, we obtain the following
expression for the sum (4.37):
(4.39)
Fp(1)+Fp(2)+Fp(3)+...+Fp(n)=Fp(n+p+1)1.
The formula (4.39) includes a further number of remarkable formulas of
discrete mathematics. In fact, for the case p=0 this formula is reduced to the
following wellknown formula for binary numbers:
20+21+22+…+2n1=2n1.
For the case p=1, the Fibonacci рnumbers coincide with the classical Fi
bonacci numbers, that is, F1(n)=Fn. And then the formula (4.39) is reduced to
the following formula:
F1+F2+F3+...+Fn=Fn+21,
that is well known from Fibonacci number theory [13, 16].
Thus, our “manipulations” with Pascal’s Triangle resulted in a small math
ematical discovery! We found an infinite number of the new numerical se
quences named the Fibonacci pnumbers (p=0,1,2,3,…). These numerical se
quences include the binary numbers (4.8) (p=0) and the classical Fibonacci
numbers (4.9) (p=1) as partial cases. These numerical sequences possess a
number of interesting mathematical properties, and their study can result in
widening the Fibonacci number theory.
4.3.7. The Ratio of Adjacent Fibonacci pNumbers
In Chapter 2 we found that the classical Fibonacci numbers are closely
connected with the golden mean. In particular, the limit of the ratio Fn /Fn1
aims for the golden mean. There is a question: what is a limit of the ratio of the
two adjacent Fibonacci pnumbers? Introduce the following definition:
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
199
Fp (n)
lim
n→∞ F (n − 1)
p
= x.
(4.40)
By using the recursive relation (4.18), we can represent the ratio of the
two adjacent Fibonacci pnumbers as follows:
Fp (n)
Fp (n − 1)
=1+
=
Fp (n − 1) + Fp (n − p − 1)
Fp (n − 1)
1
Fp (n − 1)
Fp (n − p − 1)
1
Fp (n − 1) ⋅ Fp (n − 2) ⋅⋅⋅ Fp (n − p)
=1+
.
(4.41)
Fp (n − 2) ⋅ Fp (n − 3) " Fp (n − p − 1)
Taking into consideration the definition (4.40), for the case n→∞ we can
replace the expression (4.41) by the following algebraic equation:
xp+1=xp+1.
(4.42)
Note that the equation (4.42) is called the Characteristic Equation for the
Recursive Relation (4.18).
Denote by τp a positive root of the characteristic equation (4.42). Let us
examine Eq. (4.42) for the different values of p. For p=0, Eq. (4.42) is reduced
to the trivial case: x=2. For p=1, Eq. (4.42) is reduced to the classical golden
algebraic equation:
x2=x+1
(4.43)
with a positive root τ = 1 + 5 2.
Thus, Eq. (4.42) can be considered to be a very broad generalization of the
golden equation (4.43).
(
)
4.4. The Generalized Golden pSections
4.4.1. A Generalization of the Division in Extreme and Mean Ratio
(DEMR)
Equation (4.42) has the following geometric interpretation. Let us give
the integer р a nonnegative value (p=0,1,2,3,…) and divide the line АВ at the
point С in the following proportion (Fig. 4.6):
p
CB  AB 
.
=
AC  CB 
(4.44)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
200
C
a) p = 0 A
C
b) p = 1 A
e) p = 4 A
B τ1=1.618
C
c) p = 2 A
d) p = 3 A
B τ0=2
B τ2=1.465
C
B τ3=1.380
C
B τ4=1.324
Figure 4.6. The Generalized Golden pSections
Note that the proportion (4.44) is reduced to the “dichotomy” for the case
p=0 (Fig. 4.6a) and to the classical “division in the extreme and mean ratio”
(the golden section) for the case p=1 (Fig. 4.6b). Taking this fact into consid
eration, we will name the division of the line segment AB at the point C in the
proportion (4.44) a Golden pSection and the positive root of Eq. (4.42) a Gold
en pProportion [20].
4.4.2. Algebraic Properties of the Golden рProportions
If we substitute the golden рproportion τp for x in Eq. (4.42), we get the
following identity for the golden рproportion:
τ pp +1 = τ pp + 1.
(4.45)
If we divide all terms of the identity (4.45) by τ pp , we get the following
remarkable property of the golden рproportion:
1
τp = 1+ p
τp
(4.46)
or
τp −1 =
1
.
τ pp
(4.47)
Note that for the case p=0 (τp=2) the identities (4.46) and (4.47) are
reduced to the following trivial cases:
1
1
2 = 1 + or 2 − 1 = .
1
1
(
)
For the case p=1 we have τ1 = τ = 1 + 5 2 and the identities (4.46),
(4.47) are reduced to the identities (1.11) and (1.12).
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
201
If we multiply and divide repeatedly all terms of the identity (4.45) by τp,
we obtain the following remarkable identity connecting the powers of the gold
en pproportion:
τnp = τnp−1 + τnp− p −1 = τ p × τnp−1 .
(4.48)
Note that for the case p=0 the identity (4.48) is reduced to the following
trivial identity for the binary numbers:
2n=2n1+2n1=2×2n1.
For the case p=1 the identity (4.48) is reduced to the following wellknown
identity for the classical golden mean:
τn=τn1+τn2=τ×τn1.
(4.49)
4.4.3. Geometric Progressions Based on the Golden pProportions
Now, consider a geometric progression based on the golden pproportion:
{τ , τ , ..., τ , τ , τ , τ = 1, τ , τ , τ , ...}.
n
p
n −1
p
3
p
2
p
1
p
−1
p
0
p
−2
p
−3
p
(4.50)
The geometric progression (4.50) possesses the remarkable property: for
the case p>0 according to (4.48) each term of the geometric progression (4.50),
for example, τnp can be obtained from the preceding terms in two ways: (1) by
the multiplication of the preceding term by τp τnp = τ p × τnp−1 ; (2) by the
summation of the (n1)th and the (np1)th terms τnp = τnp−1 + τnp− p −1 .
Note that till now we believed that only the golden geometric progression
based on the classical golden mean possesses similar “additive” property. It
follows from this consideration that a number of similar geometric progres
sions are infinite and all of them are based on the golden pproportions.
By decomposing τnp and all the cases arising from such decomposition terms
τnp−1 , τnp−2 , τnp− 3 , ... according to the recursive relation (4.48), we obtain the fol
lowing identities:
τnp = τnp− p −1 + τnp−1
(
(
)
)
τnp = τnp− p −1 + τnp− p −2 + τnp−2
τnp = τnp− p −1 + τnp− p −2 + τnp− p −3 + τ −p3
...
(4.51)
 k

n
n− p− j
n−k
τp =  τp
 + τp .
 j =1



In particular, for the case k=p the identity (4.51) takes the following form:
p
 p

τnp =  τnp− p − j  + τnp− p = τnp− p − j .
(4.52)
 j =1

=
0
j


∑
∑
∑
Alexey Stakhov
THE MATHEMATICS OF HARMONY
202
By decomposing τnp and terms τnp−( p +1) , τnp−2( p +1) , τnp−( k −1)( p +1) ,... arising from
such decomposition according to the recursive relation (4.48), we obtain the
following identities:
τnp = τnp−1 + τnp−( p +1)
τnp = τnp−1 + τnp−( p +1)−1 + τnp−2( p +1)
τnp = τnp−1 + τnp−( p +1)−1 + τnp−2( p+1)−1 + τ np−3( p+1)
...

 k
τ np =  τ np−( j −1)( p+1)−1  + τnp− k( p +1) .
 j =1



(4.53)
∑
In particular, for the case p=0 (τp=2) the identities (4.51) and (4.53) co
incide and they are reduced to the following remarkable identity for the “bi
nary” numbers:
k
2n =
∑2
n− j
+ 2n − k .
j =1
(4.54)
(
)
For the case p=1 we have: τ1 = τ = 1 + 5 2 ; then the identities (4.51)
and (4.53) take the following forms, respectively:
τn =
τn =
k
∑τ
n − j −1
∑τ
n − 2 j +1
j =1
k
+ τn − k
+ τn −2 k .
j =1
(4.55)
(4.56)
4.5. The Generalized Principle of the Golden Section
4.5.1. Dichotomy Principle
The remarkable book [46] by Russian architect Shevelev is devoted to a
study of the most general principles that underlie Nature. The Dichotomy Prin
ciple and the Golden Section Principle are the most important of them. The
Dichotomy Principle is based on the following trivial property of the binary
numbers:
2n=2n1+2n1,
where n=0,±1,±2,±3,… .
(4.57)
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
203
For the case n=0, we have:
1=20=21+21.
(4.58)
In the book [46] the following “dynamic” model of the Dichotomy Princi
ple is given in the form of the infinite division of the “Unit” (“The Whole”)
according to the “dichotomy” relations (4.57) and (4.58):
1=20 =21 +21
21 =22 +22
22 =23 +23
∞
(4.59)
1= 20 = 21 + 22 + 23 + 24 = ... = ∑ 2i .
i =1
4.5.2. Classical Golden Section Principle
The Golden Section Principle that came to us from Pythagoras, Plato, and
Euclid is based on the following fundamental property that connects the ad
jacent powers of the golden mean τ = 1 + 5 2 :
(
)
τ =τ +τ ,
where n=0,±1,±2,±3,… .
For the case n=1, the identity (4.60) takes the following form:
n
n1
n2
(4.60)
(4.61)
1=τ0=τ1+τ2.
Using the golden identities (4.60) and (4.81), Shevelev developed [46]
the following “dynamic” model of the Golden Section Principle:
1 = τ0 = τ −1 + τ −2
τ −2 = τ −3 + τ −4
τ −4 = τ −5 + τ −6
1 = τ0 = τ −1 + τ −3 + τ −5 + τ −7 + ... =
∞
∑τ (
− 2 i −1)
.
(4.62)
i =1
Note that the Dichotomy Principle (4.59) and the Golden Section Principle
(4.62) have a great number of applications in nature, science and mathemat
ics (binary number system, numerical methods of the algebraic equation solu
tions, self division and so on).
4.5.3. The Generalized Principle of the Golden Section
By dividing all terms of the identity (4.48) by τnp , we obtain the following
identity:
1 = τ0p = τ −p1 + τ −p p −1 .
(4.63)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
204
Using (4.48) and (4.63), we can construct the following “dynamic” model of
the “Unit” decomposition according to the Golden pProportion:
1 = τ0p = τ −p1 + τ p( )
− p +1
− p +1 −1
−2 p +1
τ p( ) = τ p( ) + τ p ( )
−2 p +1 −1
−2 p +1
−3 p +1
τp ( ) = τp ( ) + τp ( )
− p +1
1 = τ0p = τ −p1 + τ p(
− p +1)−1
+ τp (
−2 p +1) −1
+ τp (
−3 p +1)−1
+ ... =
∞
∑
− i −1 p +1 −1
τ p( )( ) .
(4.64)
i =1
The main result of the above consideration is to find more general principle
of the “Unit” division that is given by the following identity:
1 = τ −p1 + τ −p( p +1) =
∞
∑τ
− ( i −1)( p +1) −1
,
p
i =1
(4.65)
where τp is the golden pproportion, p=0,1,2,3,… .
It is clear that this general principle Generalized Principle of the Golden
Section includes the Dichotomy Principle (4.59) and the classical Golden
Section Principle (4.62) as special cases for p=0 and p=1, respectively.
4.6. A Generalization of Euclid’s Theorem II. 11
4.6.1. A Generalization of Euclid’s Theorem II.11 for the Case p=2
As we mentioned above, the golden psections that are given by the pro
portion (4.44) is a generalization of the classical golden section that is given
by (1.3). However, in Euclid’s Theorem II.11 the DEMR is formulated in the
form (1.2). We can try to represent the proportion (4.44) in the form (1.2).
We start from the partial case p=2. For this case the proportion (4.44) takes
the following form:
2
CB  AB 
(4.66)
.
=
AC  CB 
Let us denote the lengths of the line segments АВ, АС and СВ in (4.66) as
follows: AB=a, CB=b, AC=c. Then, it can be represented in the form:
(4.67)
a2×c=b3.
We can give the following geometric interpretation of the equality (4.67).
The righthand part of the equality (4.67) can be interpreted as the volume of
a cube with the side equal to b, that is, to the length of the larger segment CB
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
205
that arises at the division of a line segment АВ in the golden 2proportion (4.66).
The lefthand part of the equality (4.67) can be interpreted as the volume of a
rectangular parallelepiped. This parallelepiped has a square at its base with sides
equal to а, that is, to the length of the initial segment АВ. The height of the
rectangular parallelepiped is equal to с, that is, to the length of the smaller seg
ment АС in the proportion (4.66).
Then, taking into consideration (4.66) and (4.67), we can formulate a new
geometric problem of the division of a line in the golden 2section that is a
generalization of Euclid’s DEMR.
Generalization of DEMR (a Division in the Golden 2Proportion). Di
vide the given line АВ at point C into two segments, the smaller segment АС
and the larger segment СВ, so that a volume of the cube with the side equal to
the larger segment CB is equal to the volume of a rectangular parallelepiped
with a base, which is a square with sides equal to the initial line АВ, and with
the height equal to the smaller segment AC.
4.6.2. Euclid’s Rectangular Parallelepiped
The rectangular parallelepiped appearing in the above problem consists
of 6 faces (Fig. 4.7). The top and bottom faces are squares with sides equal to
the length of the initial segment a; the lateral faces are the rectangles with
sides equal to a and c. These rectangles are similar to Euclid’s rectangle in Fig.
4.7 where the ratio of its sides a:c for the given case is equal to the square of
the golden 2proportion τ2, that is,
a
= τ22 .
(4.68)
c
a
We will name this geometric figure Euclid’s Rect
angular Parallelepiped. Thus, according to (4.68) the
ratio of the side of its base to its height in Euclid’s Rect
c
angular Parallelepiped is equal to the square of the
golden 2proportion τ2; here, according to (4.67) its
volume is a cube of the length of the larger segment in
the proportion (4.66).
Figure 4.7.
If the initial segment AB is a unit segment (AB=1), Euclid’s rectangular
parallelepiped
then the equality (4.67) takes the following form:
c=b3.
(4.69)
Then, we can formulate the following geometric problem of the division
of the unit segment in the golden 2proportion.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
206
A Problem of the Division of the Unit Segment in the Golden 2Propor
tion. Divide a unit segment into two unequal segments such that the smaller
segment’s length is equal to the cube of the larger segment’s length.
Note that the formulated problem expresses the following property of the
golden 2proportion:
1 = τ2−1 + τ2−3 = 0.6823 + 0.3177.
(4.70)
4.6.3. The General Case of p
For the general case of р, we represent (4.44) in the following form:
a ×c=bp+1.
(4.71)
By using geometric language, we can interpret the equality (4.71) as fol
lows. The righthand part of the equality (4.71) is a volume of a hypercube in
the (p+1)dimensional space with the side equal to the length b of the larger
segment of the division of a line in the golden рproportion. The lefthand
part of the equality (4.71) is a volume of Euclid’s HyperRectangular Parallel
epiped in the (p+1)dimensional space; here, the р sides are equal to the length
а of the initial segment at the division of a line in the golden рproportion,
and the (p+1)th side (its “height”) is equal to the length с of the smaller
segment at the division of a line in the golden рproportion.
It is clear that (4.71) expresses a generalized Euclidean problem of the
division in extreme and mean ratio. We can consider this problem for the case
of the unit segment (a=1). Then, the equality (4.71) takes the following form:
p
(4.72)
c=bp+1.
Taking into consideration (4.72), we can formulate the following problem.
A Problem of the Division of the Unit Segment in the Golden pPro
portion. For a given p=0,1,2,3,… divide a unit segment into two unequal seg
ments in such proportion that the smaller segment’s length is equal to the
(p+1)th degree of the larger segment’s length.
4.7. The Roots of the Generalized Golden Algebraic Equations
4.7.1. Algebraic Equations
As is well known, algebraic equations have the following general properties,
which can be used by us before finding their roots:
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
207
1. Algebraic equations of the nth degree have n roots. In special cases it
can appear, that some roots are repeated some times (multiple roots); hence,
the number of various roots can be less than n.
2. Descartes’ “rule of signs”: the algebraic equation has no more positive
roots than the number of sign changes in a series of its factors. Often, it is
not so important to calculate the roots, it is more important to understand
the character of these roots. “The rule of signs” helps to solve this problem.
For example, the equation x54x2=0 has only one positive root because the
series of their factors 1,4,2 have only one change of a sign.
3. In the equations with real factors, the complex roots can appear only
by pairs: alongside with the root a+bi the complex number abi is always
a root of the same equation.
4. If xi (i=1,2,3,…,n) is one of the roots of the algebraic equation
(4.73)
a0xn+ a1xn1+…+an=0,
then it is easy to prove, that a polynomial that stands in the lefthand part
of Eq. (4.73), is divided by the binom (xxi) without remainder. It is easy
to prove that any polynomial of nth degree can be represented as a prod
uct of the n multipliers of the 1st degree of the kind (xxi), that is,
(4.74)
a0xn+ a1xn1+…+an=(xx1)(xx2) … (xxn).
This theorem (4.74) is sometimes named the Basic Theorem of Algebra.
4.7.2. Properties of the Roots of the Generalized Golden Algebraic
Equations
Once again, consider the characteristic equation (4.42). By using the above
rules 14, we can prove the following properties of the roots of Eq. (4.42):
1. According to Descartes’ “Rule of Signs,” Eq. (4.42) has the only positive
root τp. This root expresses some important property of Pascal Triangle.
2. As Eq. (4.42) has the degree (p+1), this means that Eq. (4.42) has (р+1)
roots x1, x2, …, xp, xp+1. Further, without loss of generality, we suppose that
the root x1 always coincides with the golden рproportion τp, that is, x1=τp.
3. As Eq. (4.42) has only the real factors 1, 1 and 1, this means that all
complex roots of Eq. (4.42) appear in pairs, that is, each complex root
a+bi appears always together with the root abi, which is the complex
conjugate to the root a+bi.
4. By using (4.74), we can represent Eq. (4.42) by its roots as follows:
xp+1xp1 =(xx1)(xx2) … (xxp)(xxp+1).
(4.75)
The general identity below for the roots x1, x2, …, xp, xp+1 comes from (4.42):
THE MATHEMATICS OF HARMONY
Alexey Stakhov
208
(4.76)
x kn = x kn −1 + x kn − p −1 = x k × x kn −1 ,
where n=0,±1,±2,±3,… and xk (k=1,2,3,…, p+1) is a root of Eq. (4.42). Note
that the identity (4.48) is a partial case of the identity (4.76) for the case k=1.
Theorem 4.1. For a given integer p>0, the following correlations for the
roots of the characteristic equation xp+1xp1=0 are valid:
x1 + x 2 + x 3 + ... + x p + x p +1 = 1
(4.77)
( x x + x x + ... + x x ) + ( x x + ... + x x ) + ... + x x = 0
( x x x + x x x + ... + x x x ) + ( x x x + ... + x x x ) + ... + x
1
2
1
3
1
p +1
2
3
2
p +1
p
p+1
p−1 x p x p +1 = 0
...
x1 x2 " x p−2 x p−1 x p + x1 x 3 x 4 " x p−1 x p x p+1 + ... + x2 x 3 x 4 " x p−1 x p x p+1 = 0
(4.78)
x1 x2 x 3 " x p−1 x p x p+1 = (−1) .
(4.79)
1
2
3
1
3
4
1
p +1
p
2
3
4
2
p
p +1
p
Proof. Consider the representation of Eq. (4.42) in the form (4.75). If we
remove the parentheses in (4.75), then, for the even p=2k we can write:
x p+1 − x p −1 = ( x − x1 )( x − x2 )"( x − x p+1 ) = x p+1 −( x1 + x2 + ... + x p+1 ) x p
+( x1 x2 + ... + x1 x p+1 + x2 x 3 + ... + x2 x p+1 + ... + x p−1 x p+1 + x p x p+1 ) x p−1
−( x1 x2 x 3 + ... + x1 x p x p+1 + x2 x 3 x 4 + ... + x2 x p x p+1 + ... + x p−1 x p x p+1 ) x p−2
+( x1 x2 x 3 x 4 + x1 x2 x 3 x5 + ... + x p−2 x p−1 x p x p+1 ) x p−3
...
+( x1 x2 " x p−1 x p + x1 x3 " x p x p+1 + ... + x2 x3 " x p x p+1 ) x − x1 x2 " x p x p+1 = 0.
(4.80)
The following outcomes follow from the comparison of (4.42) and (4.80):
x1 + x2 + ... + x p + x p+1 = 1
+( x x + x x )+ x x
( x x + ... + x x ) + ( x x + ... + x x )+ ...+
( x x x + ... + x x x ) + ( x x x + .... + x x x ) + ... + x x x = 0
1
2
1
2
1
3
p +1
1
p
2
3
p +1
2
2
3
p−1
p +1
2
4
p
p
p +1
p−1
p +1
p−1
p
p
p +1
=0
p +1
(4.81)
...
x1 x2 " x p−2 x p−1 x p + x1 x3 x4 " x p−1 x p x p+1 + ... + x2 x 3 x 4 " x p−1 x p x p+1 = 0
x1 x2 " x p x p+1 = 1.
For the odd p=2k+1, we have:
x p+1 − x p −1 = ( x − x1 )( x − x2 )"( x − x p+1 ) = x p+1 −( x1 + x2 + …+ x p+1 ) x p
+( x1 x2 + … + x1 x p+1 + x2 x3 + … + x2 x p+1 + …+ x p−1 x p + x p−1 x p+1 + x p x p+1 ) x p−1
−( x1 x2 x 3 + … + x1 x p x p+1 + x2 x 3 x 4 + …+ x2 x p x p+1 + …+ x p−1 x p x p+1 ) x p−2
+( x1 x2 x 3 x 4 + x1 x2 x 3 x5 + …+ x p−2 x p−1 x p x p+1 ) x p−3
...
−( x1 x2 x 3 " x p−2 x p−1 x p + … + x2 x3 x4 " x p−1 x p x p+1 ) x + x1 x2 x3 " x p−1 x p x p+1 = 0.
(4.82)
The following outcomes follow from the comparison of (4.42) and (4.82):
x1 + x2 + ... + x p + x p+1 = 1
+( x x + x x )+ x x
( x x + ... + x x ) + ( x x + ... + x x )+ ...+
( x x x + ... + x x x ) + ( x x x + .... + x x x ) + ... + x x x = 0
1
2
1
2
1
3
p +1
1
p
2
p +1
3
2
2
3
4
p−1
p +1
2
p
p +1
p
p−1
p +1
p−1
p
p
p +1
=0
p +1
...
x1 x2 " x p−2 x p−1 x p + x1 x3 x4 " x p−1 x p x p+1 + ... + x2 x 3 x 4 " x p−1 x p x p+1 = 0
x1 x2 " x p x p+1 = −1.
(4.83)
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
209
The outcomes (4.81) and (4.83) prove Theorem 4.1.
It is evident from (4.77) that the sum of the roots of Eq. (4.42) is identi
cally equal to 1. The expression (4.78) gives the values for every possible sum
of the roots of Eq. (4.42) taken by two, three, ..., or р roots from the (p+1)
roots of Eq. (4.42). According to (4.78), each of these sums is identically equal
to zero! At last, the expression (4.79) gives the value of the product of all
roots of Eq. (4.42). According to (4.79) this product is equal to 1 (for the even
р) or 1 (for the odd р).
Theorem 4.2. For a given integer p=1,2,3,… and for the condition when k
takes its values from the set {1,2,3,…,p}, the following identity is valid for the
roots of the characteristic equation xp+1xp1=0:
( x + x + x + ... + x ) = x + x + x + ... + x
k
1
2
3
p +1
k
1
k
2
k
3
k
p +1 = 1.
(4.84)
Proof. Consider the expression:
(x1+ x2+…+ xp+ xp+1)k,
where k takes its values from (4.86)
k∈{1,2,…,p}.
Taking into consideration the identity (4.77), we can write:
(4.85)
(4.86)
(4.87)
(x1+ x2+…+ xp+ xp+1)k =1k=1.
Consider a partial case of the expression (4.87) for the case p=1. Taking
into consideration the condition (4.86) for the case p=1 the expression (4.85)
can take the only form:
(4.88)
x1+x2.
According to the property (4.87), we can write:
(4.89)
x1+x2 = 1
that satisfies the expression (4.84).
Now, consider the case p=2. Taking into consideration the condition (4.86),
for the case p=2 the expression (4.85) can take only two different forms:
x1+x2+x3
and
(4.90)
(4.91)
(x1+x2+x3)2.
Taking into consideration the identity (4.87), we can write for the case
(4.90) the following identity:
x1+x2+x2 = 1.
Now, consider the case (4.91). Representing (4.91) in the form
(4.92)
[(x1+x2)+ x3]2,
(4.93)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
210
we can write:
( x1 + x 2 + x 3 )2 = [( x1 + x 2 ) + x 3 ] = x12 + x 22 + x 32 + 2 ( x1 x 2 + x1 x 3 + x 2 x 3 ) .
2
(4.94)
If we take into consideration the properties (4.78) and (4.84), we can re
write the expression (4.94) as follows:
( x1 + x2 + x3 )2 = x12 + x22 + x32 = 1.
(4.95)
By analogy, we can write 3 possible expressions like (4.85) for the case p=3:
x1+x2+x3+x4,
(4.96)
(x1+x2+x3+x4)2,
(4.97)
(x1+x2+x3+x4)3.
(4.98)
If we fulfill simple transformations of the expressions (4.96)(4.98) and
take into consideration the properties (4.77) and (4.84), we can write 3 iden
tities for the case p=3:
x1+x2+x3+x4 = 1
(4.99)
( x1 + x2 + x3 + x4 )2 = x12 + x22 + x32 + x42 = 1
3
( x1 + x2 + x3 + x4 ) = x13 + x23 + x33 + x43 = 1.
(4.100)
(4.101)
In the generalized case, for the proof of Theorem 4.2 for arbitrary p we can
use the socalled Multinomial Theorem [35], which is a generalization of the
binomial theorem (4.4). For any positive integer m and any nonnegative in
teger n, the multinomial formula is the following:
n

 k1 k2 k3 km
( x1 + x2 + x3 + ... + x m )n = ∑ 
 x1 x2 x 3 ... x m .
(4.102)
k1 , k2 , k3 ,..., km  k1 , k2 , k3 ,..., km 
The summation is taken over all sequences of the nonnegative integer in
m
dexes k1 through km such that ∑ ki = n . The numbers
i =1
n


n!
 k , k , k ,..., k  =
(4.103)
k1 ! k2 ! k3 !...km !
m
 1 2 3
are called Multinomial Coefficients.
Note that the Binomial Theorem (4.4) and the Binomial Coefficients (4.2)
are special cases (m=2) of the Multinomial Formula (4.102) and the Multino
mial Coefficients (4.103), respectively.
In the general case of p, the expression (4.85) can be factorized if we use
the Multinomial Formula (4.102). As is known [35], for a given k, the Multino
mial Formula (4.102) will include in itself the sum of all kth powers of Eq.
(4.42) that are taken with the coefficient of 1, that is,
x1k + x2k + x3k + ... + x pk + x pk +1 ,
(4.104)
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
211
and the sum of the products of every possible combination of two (k=2), three
(k=3) or the k roots of Eq. (4.42) that are taken with the factors known as
Multinomial Coefficients (4.103) [35]. According to Theorem 4.1 all these sums
are identically equal to zero. And then taking into consideration (4.86), we
can write the general identity (4.84).
4.7.3. Some Corollaries of Theorems 4.1 and 4.2
Now, consider some corollaries of Theorems 4.1 and 4.2 for the different val
ues of р. It is well known that for the case p=1 the golden algebraic equation
(4.43) has two real roots:
1+ 5
1 1− 5
.
and x2 = − =
τ
2
2
Hence, from the above we can obtain the following identities correspond
ing to Theorem 4.1 for the roots x1 and x2:
x1 = τ =
(4.105)
x1+x2=1; x1×x2=1.
For the case p=2, the golden algebraic equation (4.42) takes the following
form:
x3=x2+1.
(4.106)
Equation (4.106) has three roots one real root x1 and two complex con
jugate roots x2 and x3 that are given below:
h 2 1
x1 = +
+ = 1.4655712319...
(4.107)
6 3h 3
x2 = −
1 1
3 h 2
h
−
+ −i
( − ) = −0.233... − (0.793...)i
12 3h 3
2 6 3h
(4.108)
x3 = −
h
1 1
3 h 2
−
+ +i
( − ) = −0.233... + (0.793...)i ,
12 3h 3
2 6 3h
(4.109)
where
(4.110)
h = 3 116 + 12
93 .
Using direct substitution for the roots x1, x2 and x3, it is easy to prove the
following identities corresponding to Theorems 4.1 and 4.2 for the case p=2:
x1 + x 2 + x 3 = 1,
x1 x 2 + x1 x 3 + x 2 x 3 = 0,
x1 × x 2 × x 3 = 1,
x12 + x 22 + x 32 = 1.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
212
4.8. The Generalized Golden Algebraic Equations of Higher Degrees
4.8.1. The Case of p=1
In Chapter 1 we developed a theory of the golden algebraic equations of higher
degrees. We proved that for the simplest golden algebraic equation (4.43) – which
is a partial case (p=1) of the characteristic equation (4.42) there is an infinite
number of the characteristic equations with the degree greater than 2 that have
(
)
the golden mean τ = 1 + 5 2 as their general root. In their general form these
golden equations are given by the expression:
xn=Fnx2Fn2= FnxFn1,
(4.111)
where Fn, Fn1, Fn2 are Fibonacci numbers.
As we mentioned in Chapter 1, the equation x4=3x+2 which is a partial
case of the general equation (4.111) corresponding to n=4 describes the
energy state of the butadiene molecule, a valuable chemical substance, which
is used in the production of rubber. This fact at once places our interest in the
golden equations (4.111), because they, probably, in general, can describe the
energy conditions of the molecules of other chemical substances.
In this connection it is of great interest to study the characteristic equa
tions given by (4.42) and the algebraic equations following from them with
degrees more than p+1 which have the golden pproportions τp as their gen
eral roots. Let us demonstrate our approach for the partial cases p=2,3.
4.8.2. The Case of p=2
For the case of p=2 Eq. (4.42) is reduced to the algebraic equation of the
third degree: x3=x2+1. Multiplying repeatedly all terms of the equation x3=x2+1
by x, we can find the following equality for the case of p=2:
(4.112)
xn=xn1+ xn3,
where n=3, 4, 5,… .
Using the equality (4.112) and (4.42), we derive the following algebraic
equations that have the golden 2proportion τ2 as a root:
x4 = x3 + x = x2 + x + 1
x 5 = x 4 + x 2 = 2x 2 + x + 1
x 6 = x 5 + x 3 = 3x 2 + x + 2
...
x n = x n −1 + x n −3 = F2 ( n − 1) x 2 + F2 ( n − 3) x + F2 ( n − 2) ,
(4.113)
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
213
where F2(n1), F2(n2), F2(n3) are the Fibonacci 2numbers that are given by
the recursive relation (4.14) at the seeds (4.15).
As the golden 2proportion τ2 is the root of each of the equations that are
given by (4.113), the following general identity, which connects the golden 2
proportion τ2 with the Fibonacci 2numbers F2(n), follows from (4.113):
τ2n = τ2n −1 + τnn −− 33 = F2 ( n − 1) τ22 + F2 ( n − 3 ) τ2 + F2 ( n − 2) .
(4.114)
4.8.3. The Case of p=3
Consider the characteristic equation (4.42) for the case of p=3:
x =x3+1.
(4.115)
If we use the reasoning similar to the case of p=2, we can obtain the following
algebraic equations that have the golden 3proportion τ3 as their general root:
4
x5 = x4 + x = x3 + x + 1
x6 = x5 + x2 = x3 + x2 + x + 1
x 7 = x 6 + x 3 = 2x 3 + x 2 + x + 1
(4.116)
...
n
n −1
n −4
3
2
x = x + x = F3 ( n − 2 ) x + F3 ( n − 5 ) x + F3 ( n − 4 ) x + F3 ( n − 3 ) ,
where F3(n2), F3(n3), F3(n4), F3(n5) are the Fibonacci 3numbers given
by the recursive relation F3(n)=F3(n1)+F3(n4) at the seeds:
F3(1)=F3(2)=F3(3)=F3(4)=1.
As the golden 3proportion τ3 is the root of any of the equations (4.116),
the following identity, which connects the golden 3proportion τ3 with the
Fibonacci 3numbers F3(n), follows from (4.116):
τ3n = τ3n −1 + τ3n − 4 = F3 ( n − 2) τ33 + F3 ( n − 5) τ23 + F3 ( n − 4 ) τ3 + F3 ( n − 3 ) . (4.117)
4.8.4. The General Case
For the general case of р, we can write the following equality, which can
be obtained from the algebraic equation (4.42):
xn=xn1+ xnp1,
(4.118)
where n=p+1, p+2, p+3, … .
Using Eqs. (4.42) and (4.118), we can obtain the following formula for
the generalized characteristic equations that have the golden pproportion τp
as their root:
x n = Fp ( n − p + 1) x p +
p −1
∑ F (n − p − t ) x ,
t
p
t =0
(4.119)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
214
where n=p+1, p+2, p+3, …; Fp(np+1) and Fp(npt) are the Fibonacci рnum
bers given by the recursive relation (4.18) at the seeds (4.19).
It is clear that, for a given p>0, the formula (4.119) sets an infinite num
ber of generalized characteristic equations with the general root τp. The fol
lowing general identity that comes from (4.119) connects the degrees of the
golden рproportion τp with the Fibonacci рnumbers Fp(n):
τnp = Fp ( n − p + 1) τ pp +
p −1
∑ F (n − p − t ) τ ,
p
t
p
t =0
(4.120)
where n=p+1, p+2, p+3, … .
Concluding this Section we may note that the above algebraic equations
that are given by (4.111), (4.113), (4.116), and (4.119) are unusual algebraic
equations. First of all, they follow from Pascal Triangle and express some im
portant mathematical properties. Besides, they describe some harmonious
chemical and physical structures and we may expect their application for sim
ulation of many physical and chemical processes and structures.
4.9. The Generalized Binet Formula for the Fibonacci рNumbers
4.9.1. A General Approach to the Synthesis of the Generalized Binet
Formulas
In Chapter 2 we established the Binet formulas (2.62) and (2.63). These
formulas are the representations of the “extended” Fibonacci and Lucas num
bers by the golden mean τ. In this Section we try to develop a general approach
to the synthesis of Binet formulas based on the representation of the “extended”
Fibonacci pnumbers by the roots of the characteristic equation (4.42).
Our main hypothesis is the following. For a given p>0, we can represent
the Binet formula for the Fibonacci рnumbers as follows:
n
n
n
Fp ( n ) = k1 ( x1 ) + k2 ( x2 ) + ... + kp +1 ( x p +1 ) ,
(4.121)
where x1, x2, …, xp+1 are the roots of Eq. (4.42), and k1, k2, …, kp+1 are constant
coefficients that depend on the initial elements of the Fibonacci pseries.
It follows from (4.29) that Fp(0)=0 for any p>0. Therefore, we can calculate
the Fibonacci рnumbers according to the recursive relation (4.18) at the seeds:
Fp(0)=0, Fp(1)=Fp(2)=Fp(p)=1.
(4.122)
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
215
Taking into consideration (4.121) and (4.122), we can write the following
system of algebraic equations:
Fp ( 0 ) = k1 + k2 + ... + kp +1 = 0
Fp (1) = k1 x1 + k2 x2 + ... + kp+1 = 1
Fp ( 2) = k1 ( x1 ) + k2 ( x2 ) + ... + kp +1 ( x p +1 ) = 1
...
p
p
p
Fp ( p ) = k1 ( x1 ) + k2 ( x2 ) + ... + kp +1 ( x p +1 ) = 1.
2
2
2
(4.123)
Solving the system of the equations (4.123), we obtain the numerical val
ues of the coefficients k1, k2, …, kp+1 for the different values of p.
4.9.2. A Derivation of Binet Formulas for the Classical Fibonacci and
Lucas Numbers
We can use the general formula (4.121) to obtain Binet formulas for the
case p=1. For this case, the characteristic equation (4.42) is reduced to Eq.
(4.43), which has two roots x1=τ and x2=1/τ, where τ = 1 + 5 2.
Therefore, formula (4.121), for the case of p=1, takes the following form:
n
n
 1
F1 ( n ) = k1 ( τ ) + k2  −  .
(4.124)
 τ
It is clear that for the case of p=1 the system of algebraic equations (4.123) is
 F1 ( 0 ) = k1 + k2 = 0

(4.125)
 F1 (1) = k1τ + k2 ( − 1 τ ) = 1.
By solving the system (4.125), we obtain: k1 = 1 5 and k1 = − 1 5 . If we
substitute k1 and k2 into (4.124), we obtain the wellknown Binet formula for
the classical Fibonacci numbers in the form:
n
τn − ( − 1 τ )
.
F1 (n) =
(4.126)
5
If we assume k1=k2=1 in (4.124), we obtain the Binet formula for the clas
sical Lucas numbers:
(
)
(4.127)
L1 (n) = τn + ( − 1 τ ) .
This formula generates the Lucas series at the seeds L1(0)=2 and L1(1)=1:
n
2, 1, 3, 4, 7, 11, 18, 29, … .
(4.128)
Let us note one important fact regarding the seed F1(0)=0. It follows di
rectly from the Binet formula (4.124). In fact, according to (4.124) we have
the following result for the case of n=0:
Alexey Stakhov
THE MATHEMATICS OF HARMONY
216
(4.129)
F1 ( 0 ) = k1τ0 + k2 ( − 1 τ ) = k1 + k2 = 1 5 − 1 5 = 0.
These simple calculations show that the seed F1 ( 0 ) = 0 comes from the
fact that the sum of the coefficients k1 and k2 in the expression (4.125) is also
equal to zero.
0
4.9.3. Binet Formula for the Fibonacci 2Numbers
Let us give p=2 and use the above approach for obtaining Binet formula
for the Fibonacci 2numbers. For the case p=2, the recursive relation for the
Fibonacci 2numbers and the characteristic equation (4.42) take the follow
ing forms, respectively:
F2(n)=F2(n1)+F2(n3)
(4.130)
F2(0)=0, F2(1)=F2(2)=1
(4.131)
x3=x2+1.
(4.132)
Equation (4.132) has three roots real (positive) root x1=τ2 given by
(4.107) and two complexconjugate roots x2 and x3 (4.108) and (4.109).
We recall that the real root x1 of the algebraic equation (4.132) is an irra
tional number that is equal to the golden 2proportion (x1=τ2). The number h
given by (4.110) is also irrational; hence, the roots x2 and x3 are complex num
bers with irrational real parts.
For the case p=2, the formula (4.121) and the system (4.123) take the
following forms, respectively:
n
n
n
F2 ( n ) = k1 ( x1 ) + k2 ( x2 ) + k3 ( x3 )
(4.133)
F2 ( 0 ) = k1 + k2 + k3 = 0
F2 (1) = k1 x1 + k2 x2 + k3 x 3 = 1
F2 ( 2) = k1 ( x1 ) + k2 ( x2 ) + k3 ( x3 ) = 1.
2
2
2
(4.134)
Solving the system (4.134), we obtain:
k1 =
2h ( h + 2 )
( h + 8)
(4.135)
3
 − ( h + 2) + i 3 ( h − 2)  h

k2 = 
3
h +8
(
)
(4.136)
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
217
 − ( h + 2) − i 3 ( h − 2)  h
 ,
k3 = 
h3 + 8
(
)
(4.137)
where h is given by (4.110).
Note that the coefficient (4.135) is an irrational number because the num
ber h given by (4.110) is irrational. Also the coefficients (4.136) and (4.137)
are complexconjugate numbers with irrational real parts.
Substituting (4.107)(4.110) and (4.135)(4.137) into the expression
(4.133), we can write the following Binet formula for the Fibonacci 2
numbers, which is a generalization of the Binet formula (4.126) for the classi
cal Fibonacci numbers :
F2 ( n ) =
2h ( h + 2 )  h 2 1 
+
+
h3 + 8  6 3h 3 
n
n
 − ( h + 2) + i 3 ( h − 2) h  h
1 1
3  h 2 


+
+ −i
−
− −

2  6 3h  
h3 + 8
 12 3h 3
n
 − ( h + 2 ) − i 3 ( h − 2 ) h  h
1 1
3  h 2 


+ +i
+
 − −
 6 − 3h   .
12
3
3
2
h
h3 + 8



(4.138)
It seems incredible at first sight, that the formula (4.138) that is
very complicated combination of complex numbers with irrational coeffi
cients represents the integer Fibonacci 2numbers F 2(n) for any integer
n=0, ±1, ±2, ±3, ... .
4.9.4. Binet Formula for the Fibonacci 3Numbers
For the case of p=3, the recursive relation for the Fibonacci 3numbers and
the characteristic equation (4.42) take the following forms, respectively:
F3(n)=F3(n1)+F3(n4)
(4.139)
F3(0)=0, F3(1)=F3(2)=F3(3)=1
(4.140)
x4=x3+1.
(4.141)
Equation (4.141) has four roots two real roots, x1 and x2, and two com
plexconjugate roots, x3 and x4. The roots of Eq. (4.141) are irrational and
complex numbers, which have a very complex symbolic representation; there
fore, we can use their approximate numerical values:
x1=1.380; x2=0.819; x3=0.219+0.914i; x4=0.2190.914i.
(4.142)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
218
It is important to note that the root x1 of Eq. (4.141) is the golden 3propor
tion (x1=τ3=1.380).
The formulas (4.122) for the Fibonacci 3numbers and the system of the
algebraic equations (4.123) take the following forms, respectively:
F3 ( n ) = k1 ( x1 ) + k2 ( x2 ) + k3 ( x3 ) + k4 ( x4 )
n
n
n
n
(4.143)
F3 ( 0 ) = k1 + k2 + k3 + k4 = 0
F3 (1) = k1 x1 + k2 x2 + k3 x3 + k4 x4 = 1
F3 ( 2) = k1 ( x1 ) + k2 ( x2 ) + k3 ( x3 ) + k4 ( x4 ) = 1
2
2
2
2
(4.144)
F3 ( 3 ) = k1 ( x1 ) + k2 ( x2 ) + k3 ( x3 ) + k4 ( x4 ) = 1.
3
3
3
3
Solving the system (4.144), we obtain the following numerical coefficients:
(4.145)
k1=0.3969; k2=0.1592; k3=0.11880.2045i; k4=0.1188+0.2045i.
Hence, using (4.142) and (4.145), we can represent Binet formula (4.143)
for the Fibonacci 3numbers in the following numerical form:
F3 ( n ) = 0.3969 (1.38 ) − 0.1592 ( −0.819 )
n
+ ( −0.1188 − 0.2045i ) ( 0.219 + 0.914i )
n
+ ( −0.1188 + 0.2045i ) ( 0.219 − 0.914i ) .
n
n
(4.146)
4.9.5. Binet Formulas for the Fibonacci 4Numbers
For the case of p=4, the recursive relation for the Fibonacci 4numbers and
the characteristic equation (4.42) take the following forms, respectively:
F4(n)= F4(n1)+F4(n5)
(4.147)
F4(0)=0, F4(1)=F4(2)= F4(3)= F4(4)=1
(4.148)
x5=x4+1.
(4.149)
Equation (4.141) has five roots one real root x1 that coincides with the gold
en 4proportion τ4 and two pairs of complexconjugate roots x2, x3 and
x4, x5. They all can be represented in the following analytical and numerical form:
x1 =
h 2
1 i 3
1 i 3
+ = 1.3247, x2 = −
= 0.5 − 0.866i, x 3 = +
= 0.5 + 0.866i
6 h
2
2
2
2
 h 1 i 3 h 2
x4 = −  +  −
 −  = −0.6623 − 0.5623i
 12 h  2  6 h 
 h 1 i 3 h 2
x5 = −  +  +
 −  = −0.6623 + 0.5623i,
 12 h  2  6 h 
(4.150)
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
219
where h = 3 108 + 12 69 .
For this case, the formula (4.121) for the Fibonacci 4numbers and the
system of the algebraic equations (4.123) take the following forms:
n
n
n
n
n
F4 ( n ) = k1 ( x1 ) + k2 ( x2 ) + k3 ( x3 ) + k4 ( x4 ) + k5 ( x5 )
(4.151)
F4 ( 0 ) = k1 + k2 + k3 + k4 + k5 = 0
F4 (1) = k1 x1 + k2 x2 + k3 x 3 + k4 x 4 + k5 x5 = 1
F4 ( 2 ) = k1 ( x1 ) + k2 ( x2 ) + k3 ( x3 ) + k4 ( x4 ) + k5 ( x5 ) = 1
2
2
2
2
2
F4 ( 3 ) = k1 ( x1 ) + k2 ( x2 ) + k3 ( x 3 ) + k4 ( x 4 ) + k5 ( x5 ) = 1
3
3
3
3
3
(4.152)
F4 ( 4 ) = k1 ( x1 ) + k2 ( x2 ) + k3 ( x 3 ) + k4 ( x 4 ) + k5 ( x5 ) = 1.
Solving the system (4.152), we obtain the numerical values of the coefficients:
4
4
4
4
k1 = 0.380; k2 = −0.171 + 0.206i; k3 = −0.071 − 0.206i;
k4 = −0.119 + 0.046i; k5 = −0.119 − 0.046i.
4
(4.153)
Using (4.150) and (4.153), we can represent the Binet formula for the
Fibonacci 4numbers (4.151) in numerical form.
4.9.6. A General Case of p
In the general case, Binet formula for the Fibonacci рnumbers has the
form (4.121). The coefficients k1, k2, …, kp+1 in the formula (4.121) are the solu
tions of the system (4.123). This outcome can be formulated as the following
theorem.
Theorem 4.3 (Generalized Binet Formula for the Fibonacci p
Numbers). For a given integer p>0, any Fibonacci рnumber Fp(n) (n=0, ±1,
±2, ±3, …) given by the recursive relation Fp(n)=Fp(n1)+Fp(np1) at the seeds
Fp(0)=0, Fp(1)=Fp(2)= … =Fp(p)=1 can be represented in the following ana
lytical form:
Fp ( n ) = k1 ( x1 ) + k2 ( x2 ) + ... + kp +1 ( x p +1 ) ,
where x1, x2, …, xp+1 are the roots of the characteristic equation xp+1=xp+1 and
k1, k2, …, kp+1 are constant coefficients that are the solutions of the system of
the algebraic equations:
n
n
n
Fp ( 0 ) = k1 + k2 + ... + kp +1 = 0
Fp (1) = k1 x1 + k2 x2 + ... + kp +1 x p +1 = 1
(
)
2
Fp ( 2 ) = k1 ( x1 ) + k2 ( x2 ) + ... + kp +1 x p +1 = 1
...
p
p
p
Fp ( p ) = k1 ( x1 ) + k2 ( x2 ) + ... + kp +1 x p +1 = 1.
2
2
(
)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
220
Proof. Represent the Fibonacci рnumber Fp(p+1) in the analytical form
according to (4.121):
p +1
p +1
p +1
Fp ( p + 1) = k1 ( x1 ) + k2 ( x2 ) + ... + kp +1 ( x p +1 ) .
(4.154)
Using the identity (4.76), we can represent the expression (4.154) in the
following form:
p
p
p
F p ( p + 1) =  k1 ( x1 ) + k2 ( x 2 ) + ... + kp +1 x p +1 


0
0
0
(4.155)
+  k1 ( x1 ) + k2 ( x 2 ) + ... + k p +1 x p +1  .


Then, using the definition (4.121), we can write:
(
(
)
)
Fp(p+1)= Fp(p)+Fp(0).
(4.156)
This means that the recursive relation (4.18) is valid for the Fibonacci
pnumber Fp(p+1), that is, the analytical formula (4.154) represents the
Fibonacci pnumber Fp(p+1).
Furthermore, applying the formula (4.121) for the Fibonacci рnumbers
Fp(p+2), Fp(p+3), …, Fp(n), … and using the identity (4.76), it is easy to prove
that the analytical formula (4.121) represents all Fibonacci pnumbers for the
positive values of n.
Let us prove that Eq. (4.121) is valid for the negative values of n=1, 2, 3, … .
In order to do this, we consider the formula (4.121) for the case n=1:
−1
−1
−1
Fp ( −1) = k1 ( x1 ) + k2 ( x2 ) + ... + kp +1 ( x p +1 ) .
(4.157)
We can rewrite the identity (4.76) in the following form:
x kn − p −1 = x kp − x kn −1 .
(4.158)
For the case n=p, the identity (4.158) takes the following form:
x k−1 = x kp − x kp −1 .
(4.159)
Using the identity (4.159), we can rewrite (4.157) as follows:
( )
( )
p
p
p
F p ( −1) =  k1 ( x1 ) + k2 ( x 2 ) + ... + kp +1 x p +1 


p −1
p −1
p −1
(4.160)


.
− k1 ( x1 ) + k2 ( x 2 ) + ... + kp +1 x p +1


Hence, using the general formula (4.121), we can see that the formula
(4.160) is equivalent to
Fp(1)= Fp(p)+Fp(p1)=11=0,
that is, the formula (4.160) represents the Fibonacci рnumber Fp(1)=0.
Furthermore, considering the formula (4.121) for the negative values of
n=2, 3, 4, … and using (4.158), it is easy to prove that the formula (4.121) is
valid for all negative values of n.
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
221
4.10. The Generalized Lucas pNumbers
4.10.1. Binet Formula for the Lucas pNumbers
Next let us generalize the Binet formula for the classical Lucas numbers that
are given by (4.127). We recall that the formula (4.127) can be obtained from
the Binet formula for Fibonacci numbers that are given by (4.126), provided we
assume in it that k1=k2. We can use this approach for the introduction of a new
class of the recursive sequences. If we assume that k1=k2=…=kp+1=1 in the for
mula (4.121), we can obtain the following formula:
Lp ( n ) = ( x1 ) + ( x2 ) + ... + ( x p +1 ) .
(4.161)
Note that for the case p=1 this formula is reduced to the Binet formula for
the classical Lucas numbers that are given by (4.127). Let us prove that this
formula represents a new class of recursive numerical sequences called Lucas
pnumbers. They are given by the following recursive relation:
n
n
n
Lp(n)= Lp(n1)+Lp(np1)
at the seeds:
(4.162)
Lp(0)=p+1
(4.163)
Lp(1)=Lp(2)=…=Lp(p)=1.
(4.164)
In fact, for the case n=0, we can write the formula (4.161) as follows:
Lp ( 0 ) = ( x1 ) + ( x2 ) + ... + ( x p +1 ) = p + 1.
This proves that for the case n=0 the formula (4.161) gives the seed (4.163).
Now, consider the formula (4.161) for the cases of n=1, 2, 3, ..., р:
0
0
0
Lp (1) = x1 + x 2 + ... + x p +1
( )
L ( 3 ) = ( x ) + ( x ) + ... + ( x )
Lp ( 2 ) = ( x1 ) + ( x 2 ) + ... + x p +1
2
2
3
p
1
3
2
3
p +1
2
...
p
p
p
Lp ( p ) = ( x1 ) + ( x 2 ) + ... + x p +1 .
(
(4.165)
)
Then, according to Theorem 4.2, all expressions (4.165) are equal to 1.
This proves that the expression (4.161) is valid for the cases n=1, 2, 3, …, p.
To prove the validity of the recursive relation (4.162) for the general case
of n, we can use the identity (4.76) and represent the formula (4.161) as follows:
Alexey Stakhov
THE MATHEMATICS OF HARMONY
222
(
)
n −1
n −1
n −1

Lp ( n ) = ( x1 ) + ( x 2 ) + ... + x p +1


n − p −1
n − p −1
n − p −1
.
+ ( x2 )
+ ... + x p +1
+ ( x1 )


(
(4.166)
)
Using the definition (4.161), we can rewrite the expression (4.166) in the
form of the recursive relation (4.162).
Our reasoning resulted in the discovery of new recursive numerical se
quences Lucas pnumbers given by the recursive relation (4.162) at the seeds
(4.163) and (4.164). It is clear that for the case p=1 this recursive relation is
reduced to the recursive relation for the classical Lucas numbers.
Let us study the partial cases of the Lucas pnumbers for the cases p=2, 3, 4.
4.10.2. Binet Formula for the Lucas 2Numbers
For the case p=2, the formula (4.161) can be presented in the form below:
L2 ( n ) = ( x1 ) + ( x2 ) + ( x3 ) .
n
n
n
(4.167)
This formula defines the Lucas 2numbers L2(n).
If we substitute the expressions for the roots x1, x2, x3, given by (4.107)
(4.110) into (4.167) we can rewrite the formula (4.167) as follows:
n
1 1 i 3  h 2 
h 2 1  h
L2 ( n ) =  +
+  + − −
+ +
−

2  6 3h  
 6 3h 3   12 3h 3
n
 h
1 1 i 3  h 2 
+ − −
+ −
−
 .
2  6 3h  
 12 3h 3
n
(4.168)
For the case p=2, the recursive relation (4.162) and the seeds (4.163) and
(4.164) are reduced to the following:
L2(n)=L2(n1)+L2(n3)
(4.169)
L2(0)=3
(4.170)
L2(1)=L2(2)=1.
(4.171)
Then, using the recursive relation (4.169) at the seeds (4.163) and (4.164),
we can calculate all elements of the Lucas 2series:
3, 1, 1, 4, 5, 6, 10, 15, 21, 31, 46, 67, 98, 144, … .
4.10.3. Binet Formula for the Lucas 3Numbers
For the case p=3, the formula (4.161) takes the following form:
(4.172)
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
223
(4.173)
L3 ( n ) = ( x1 ) + ( x2 ) + ( x3 ) + ( x4 ) .
This formula is an analytical expression for the Lucas 3numbers L3(n).
Using the numerical values for the roots x1, x2, x3, x4 given by (4.142), we
can represent the formula (4.173) in the following numerical form:
n
n
n
n
L3(n)=(1.380)n+(0.819)n+(0.219+0.914i)n+(0.2190.914i)n.
(4.174)
For the case p=3 the recursive relation (4.162) and the seeds (4.163) and
(4.164) are reduced to the following:
L3(n)= L3(n1)+ L3(n4)
(4.175)
L3(0)=4
(4.176)
(4.177)
L3(1)= L3(2)=L3(3)=1.
The recursive relation (4.175) at the seeds (4.176) and (4.177) generates
the following Lucas 3series:
4,1,1,1,5,6,7,8,13,19,26,34,47,66,… .
(4.178)
4.10.4. Binet Formula for the Lucas 4Numbers
For the case p=4, the formula (4.161) takes the following form:
n
n
n
n
n
L4 ( n ) = ( x1 ) + ( x2 ) + ( x3 ) + ( x4 ) + ( x5 ) .
(4.179)
This formula sets the Lucas 4numbers L4(n) in analytical form.
Substituting the analytical representations of the roots x1, x2, x3, x4, x5 giv
en by (4.150) into (4.179), we obtain the following formula:
h
L4 ( n ) = 
6
n
+
n
2 1 i 3  1 i 3 
+ −
 + +

2   2
2 
h   2
n
n
n
(4.180)
 h 1 i 3  h 2   h 1 i 3  h 2 
−
+
−
−
+
−
+ − − −
.



2  6 6    12 6
2  6 6  
 12 6
For the case p=4, the recursive relation (4.162) and the seeds (4.163) and
(4.164) are reduced to the following:
L4(n)= L4(n1)+ L4(n5)
(4.181)
L4(0)=5
(4.182)
L4(1)= L4(2)=L4(3)=L4(4)=1.
(4.183)
The recursive relation (4.181) at the seeds (4.182) and (4.183) generates
the following Lucas 4series:
5, 1, 1, 1, 1, 6, 7, 8, 9, 10, 16, 23, 31, 40, … .
(4.184)
THE MATHEMATICS OF HARMONY
Alexey Stakhov
224
4.10.5. The “Extended” Lucas рNumbers
Above we introduced the socalled “extended” Fibonacci pnumbers that
are given for the negative values of n (see Table 4.2). By analogy, we can in
troduce the “extended” Lucas pnumbers, if we extend the Lucas pnumbers
to the side of the negative values of n. With this purpose in mind, we shall find
some general properties of such “extended” sequences. For the calculation of
the Lucas pnumbers Lp(0), Lp(1), Lp(2),…, Lp(p),…, Lp(2p+1), …, which
correspond to the nonnegative values of n=1,2,3,…, we represent the re
cursive relation (4.162) as follows:
Lp(np1)= Lp(n)Lp(n1).
In particular, for the case n=p the formula (4.185) is
(4.185)
(4.186)
Lp(1)= Lp(p)Lp(p1).
Now, consider the formula (4.185) for the different values of p. For p=1,
the formulas (4.185) and (4.186) take the following forms, respectively:
L1(n2)=L1(n)L1(n1).
(4.187)
L1(1)=L1(1)L1(0).
As L1(1)=1 and L1(0)=2, it comes from (4.188) that
(4.188)
L1(1)=1.
(4.189)
Using the recursive relation (4.186), we can calculate all values of the
Lucas numbers L1(n) for the negative values of n=1, 2, 3, … and then repre
sent the classical Lucas numbers L1(n) as is shown in Table 4.3.
Table 4.3. The “extended” classical Lucas numbers
n
5
4
3
2
1
0
1
2
3
4
5
L1(n)
11
7
4
3
1
2
1
3
4
7
11
For the case p=2, the formulas (4.185) and (4.186) take the following form:
L2(n3)=L2(n)L2(n1)
(4.190)
L2(1)=L2(2)L2(1).
Taking into consideration (4.171), we can write:
(4.191)
(4.192)
L2(1)=0.
Using (4.170) and calculating numerical values of the Lucas 2numbers
L2(n) for the nonnegative n=0, 1, 2, 3, 4, ... , we can get the following
numerical sequence:
L2(n) (n≤0): 3, 0, 2, 3, 2, 5, 1, 7, 6, 6, … .
(4.193)
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
225
For the case p=3, the formulas (4.185) and (4.186) take the following form:
L3(n4)=L3(n)L3(n1)
(4.194)
L3(1)=L3(3)L3(2).
(4.195)
Using (4.170) and (4.194) and (4.195), we can calculate numerical values
of the Lucas 3numbers L3(n) for the nonnegative n=0, 1, 2, 3, 4, ..., we can
get the following numerical sequence L3(n) (n≤0): 4, 0, 0, 3, 4, 0, 3, 7, 4, 3, … .
Of course, by using the general recursive relation (4.185), we can calcu
late the rest of the values of Lp(n) for the nonnegative values of n.
In Table 4.4 we can see the “extended” Lucas numbers Lp(n) for the cases
p=1, 2, 3, 4.
Table 4.4. The “extended” Lucas pnumbers
n
6
5
4
3
2
1
0
1
2
3
4
5
6
L1(n)
18
11
7
4
3
1
2
1
3
4
7
11
18
L2(n)
10
6
5
4
1
1
3
0
2
3
2
5
1
L3(n)
7
6
5
1
1
1
4
0
0
3
4
0
3
L4(n)
7
6
1
1
1
1
5
0
0
0
4
5
0
4.10.6. Identities for the Sums of the Lucas рNumbers
Once again, consider the recursive relation (4.169). Decomposing the Lucas
2number L2(n1) in (4.169) according to the same recursive relation (4.169),
that is, representing L2(n1) in the form
L2(n1)=L2(n2)+L2(n4),
we can represent the formula (4.169) in the following form:
(4.196)
L2(n)=L2(n2)+ L2(n3)+L2(n4).
This means that the sum of the three sequential Lucas 2numbers is al
ways equal to the Lucas 2number that is two positions from the senior Lucas
2number of the sum.
If we use a similar approach for the Lucas рnumbers in the general case,
we obtain the following general identity:
Lp(n)=Lp(np)+Lp(np1)+Lp(np2)+…+Lp(n2p).
(4.197)
Note that the identity (4.197) is valid for all “extended” Lucas pnumbers.
Now, let us examine the sum of the first n Lucas pnumbers:
Alexey Stakhov
THE MATHEMATICS OF HARMONY
226
Lp(1)+Lp(2)+Lp(3)+…+Lp(n).
(4.198)
In order to get the required result, we write down the basic recursive rela
tion (4.162) for the Lucas pnumbers in the following form:
Lp(n)=Lp(n+p+1)Lp(n+p).
Using (4.199), we can write the following equalities:
(4.199)
Lp (1) = Lp ( 2 + p ) − Lp (1 + p )
Lp ( 2 ) = Lp ( 3 + p ) − Lp ( 2 + p )
Lp ( 3 ) = Lp ( 4 + p ) − Lp ( 3 + p )
...
Lp ( n − 1) = Lp ( n + p ) − Lp ( n + p − 1)
Lp ( n ) = Lp ( n + p + 1) − Lp ( n + p ) .
Summing term by term the lefthand and righthand parts of these equal
ities, we obtain the following formula:
Lp(1)+Lp(2)+Lp(3)+…+Lp(n)= Lp(n+p+1)Lp(1+p).
(4.200)
It follows from the recursive relation (4.162) and the seeds (4.163) and
(4.164) that
(4.201)
Lp(1+p)=Lp(p)+Lp(0)=1+p+1=p+2.
The following formula for the sum (4.200) follows from (4.201):
Lp(1)+Lp(2)+Lp(3)+…+Lp(n)= Lp(n+p+1) p2.
(4.202)
4.10.7. The Ratio of Adjacent Lucas pNumbers
Above we found that the Fibonacci pnumbers are closely connected with
the golden pproportion. In particular, the limit of the ratio Fp(n)/Fp(n1)
aims for the golden pproportion. There is a question: what is the limit of the
ratios of adjacent Lucas pnumbers? Introduce the following definition:
L p ( n)
lim
= x.
(4.203)
n →∞ L (n − 1)
p
Using the recursive relation (4.162), we can represent the ratio of the ad
jacent Lucas pnumbers as follows:
L p ( n)
Lp (n − 1)
= 1+
1
Lp (n − 1)
Lp (n − p − 1))
=
Lp (n − 1) + Lp (n − p − 1)
= 1+
Lp (n − p − 1)
1
Lp (n − 1) ⋅ Lp (n − 2) ⋅⋅⋅ Lp (n − p)
Lp (n − 2) ⋅ Lp (n − 3) " Lp (n − p − 1)
.
(4.204)
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
227
Taking into consideration the definition (4.203), for the case n→∞ we can
exchange the formula (4.204) by the algebraic equation (4.42) with the positive
root τp the golden pproportion. It follows from this reasoning that
lim
L p ( n)
n →∞ L
p (n − 1)
= τp.
(4.205)
4.11. The “Metallic Means Family” by Vera W. de Spinadel
4.11.1. The Generalized Fibonacci Sequences
Recently the Fibonacci numbers and the golden mean were generalized
by the Argentinean mathematician Vera W. de Spinadel who is the author of
the original book in this regard [42]. Spinadel generalized the Fibonacci re
cursive relation F(n+1)=F(n)+F(n1) as follows:
G(n+1)=pG(n)+qG(n1),
(4.206)
where p and q are natural numbers.
Consider the examples of the generalized Fibonacci sequences of the kind
(4.206). If we assume that p=2 and q=1 in (4.206) and begin from the seeds
G(1)=G(2)=1, then the recursive relation (4.206) generates the following
generalized Fibonacci numbers:
1, 1, 3, 7, 17, 41, 99, 140, … .
(4.207)
For the case p=3 and q=1, and G(1)=G(2)=1 the generalized Fibonacci
numbers are:
1, 1, 1 , 4, 13, 43, 142, 469, … .
(4.208)
We can represent the recursive relation (4.206) in the following form:
G(n + 1)
G(n − 1)
q
= p+q
= p+
.
G ( n)
G ( n)
G ( n)
(4.209)
G(n − 1)
If we denote by x the limit of the ratio of two adjacent generalized Fi
bonacci numbers, that is,
G(n + 1)
x = lim
,
n →∞ G(n)
then we can represent the expression (4.209) as follows:
Alexey Stakhov
THE MATHEMATICS OF HARMONY
228
x = p+
q
x
(4.210)
or
x2pxq=0.
This algebraic equation has the following positive solution:
x=
p + p 2 + 4q
.
2
This means that
G(n + 1) p + p2 + 4q
.
=
n →∞ G(n)
2
lim
(4.211)
(4.212)
(4.213)
4.11.2. The Metallic Means Family
Above we have introduced the quadratic algebraic equation (4.211). Vera
W. de Spinadel proved that Eq. (4.211) gives an infinite set of positive qua
dratic irrationals for the different values of p and q. They are all given by the
formula (4.212) and together make the Metallic Means Family (MMF).
Let us denote any member of the MMF by σqp , where p and q take their
values from the set of natural numbers, that is, p=1, 2, 3, … ; q=1, 2, 3, … .
Consider special cases of Eq. (4.211). We start from the case q=1, that is,
from the following algebraic equation:
(4.214)
x2px1=0.
It is convenient to represent the members of the MMF in the form of a
continued fraction. It is clear that for the case p=1 Eq. (4.214) is reduced to
the simplest algebraic equation
x2x1=0
with the positive root equal to the golden mean.
Equation (4.215) can be written in the form:
(4.215)
1
.
(4.216)
x
By substituting the golden mean τ for x in Eq. (4.216), we obtain the
following representation of τ :
1
τ =1+ .
(4.217)
τ
It comes from (4.217) that the following representation of the golden
x =1+
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
229
mean in the form of a continued fraction:
1
.
τ =1+
1
1+
(4.218)
1
1+
1 + ...
Note that the representation (4.218) was considered in Chapter 1.
In mathematics, for the compact representation of the continued fractions
the following representation is used:
ϕ = [1,1,1,...] =  1  .
(4.219)
Now, assume that p=2 in Eq. (4.214), that is, consider the following equation:
(4.220)
x22x1=0.
Then, according to the above definition we can denote the positive root of
Eq. (4.220) by σ12 . We can represent Eq. (4.220) in the form:
1
x =2+ .
(4.221)
x
If we substitute 2+(1/x) for x on the righthand part of (4.221), we obtain
the following representation of x:
x =2+
1
.
1
(4.222)
x
Continuing this process ad infinitum, we obtain the representation of σ12
2+
in the form of the following continued fraction:
1
.
σ12 = 2 +
1
2+
(4.223)
1
2+
2 + ...
The positive quadratic irrational number σ12 given by (4.223) was named
by Spinadel the Silver Mean. By analogy with the golden mean (4.219), the
Silver Mean (4.223) can be represented in the following compact form:
σ12 = [ 2, 2, 2,...] =  2  .
(4.224)
Using the formula (4.212), we can write the analytical representation of
the silver mean σ12 as follows:
σ12 = 1 + 2 =  2  .
(4.225)
If we assume p=3, then Eq. (4.214) takes the following form:
x 3x1=0.
2
(4.226)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
230
The positive root of this equation is called the Bronze Mean σ13 ; it can be
represented in the form of the following continued fraction:
1
σ13 = 3 +
1
3+
(4.227)
1
3+
3 + ...
or
σ12 = [ 3, 3, 3,...] =  3  .
(4.228)
Using the formula (4.212), we can write the analytical representation of
the bronze mean σ13 as follows:
3 + 13
=  3  .
(4.229)
2
For the cases where p=4, 5, 6, 7, 8, 9, 10, respectively, we can find the
following analytical representations of the corresponding Metallic Means:
σ13 =
5 + 29
7 + 53
; σ16 = 3 + 10 ; σ17 =
;
2
2
9 + 85
σ18 = 4 + 17 ; σ19 =
; σ110 = 5 + 26 .
2
It is clear that all Metallic Means of the kind σ1p have the following gen
σ14 = 2 + 5 ; σ15 =
eral representation in the form of the periodic continued fraction:
1
= [ p ].
σ1p = p +
1
p+
1
p+
p + ...
(4.230)
4.11.3. Other Types of the Metallic Means
If we assume that p=1 in Eq. (4.211), then we obtain the following equation:
(4.231)
x2xq=0,
where q is a natural number. The positive roots of this equation generate a
new class of the Metallic Means denoted by σ1q .
Note that for the case q=1, Eq. (4.231) is reduced to the golden equation
(4.215). For the case q=2, Eq. (4.231) takes the following form:
x2x2=0.
(4.232)
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
231
The number σ1q = 2 is a positive root of this equation. Spinadel calls this
number the Copper Mean. Using a traditional representation of the continued
fraction, we can represent the Copper Mean as follows:
σ1q = 2 = 2, 0  .
If we assume that q=3 in Eq. (4.231), then we obtain the following equation:
2
x x3=0.
This equation results in the Nickel Mean
(4.233)
1 + 13
= 2, 3  .
(4.234)
2
By analogy, we can find the next representations of the other Metallic
Means of the type σ1q .
σ13 =
4.11.4. PisotVijayaraghavan Numbers and Metallic Means
Vera W. de Spinadel paid attention to the fact that her Metallic Means
have a direct relation to PisotVijayaraghavan numbers or PVnumbers [152].
It is well known that the PVnumbers are positive roots of the following alge
braic equation:
(4.235)
x m = am −1 x m −1 + ... + ai x i + ... + a1 x + a0 ,
where ai are integers.
The golden mean τ = 1 + 5 2 is the example of the PVnumbers be
cause the golden mean is the positive root of Eq. (4.215), which is a partial
case of (4.235). Also, all the golden pproportions tp are PVnumbers because
Eq. (4.42) is a partial case of (4.235).
The number Q1=1.324… a positive root of the equation x3x1=0 is also
a PVnumber. This number is called a Plastic Constant.
It is proved that Eq. (4.215) with the root τ = 1 + 5 2 appears in qua
sicrystal structures. In addition, the following algebraic equations which
are partial cases of (4.235) appear in quasicrystal structures:
(
)
(
)
x 2 − 2x − 1 = 0 → γ = 1 + 2
(4.236)
(4.237)
x 2 − 4x + 1 = 0 → δ = 1 + 3.
τ
=
1
+
5
2
(the golden mean)
Note that the particular PVnumber
corresponds to pentagonal and decagonal quasilattices, while another PV
number γ = 1 + 2 (the Silver Mean) corresponds to the octagonal quasilat
tice, and the PVnumber δ = 1 + 3 corresponds to the case of the dodecago
nal quasilattice.
(
)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
232
The above examples demonstrate that Spinadel’s Metallic Means are of great
theoretical importance for algebra and crystallography.
4.11.5. “Silver Means” by Jay Kappraff
The American researcher Jay Kappraff – the author of the interesting books
[47, 50 ] – developed an approach to the generalization of Fibonacci numbers
and the golden mean similar to Spinadel’s Metallic Means. He considered the
recursive relation
G(n+1)=N G(n)+G(n1)
(4.238)
that is a partial case of the recursive relation (4.206) for the case q=1 and
p=N. This recursive relation gives different generalized Fibonacci numbers at
different seeds. From the recursive relation (4.238) Kappraff derived the fol
lowing algebraic equation
1
= N,
(4.239)
x
which is the other form of the equation (4.214).
It is clear that for the case N=1 and the seeds G(0)=0 and G(1)=1 the
recursive relation (4.238) generates the classical Fibonacci numbers 0, 1, 1, 2,
3, 5, 8, 13, 21, 34, ... . The ratios of the adjacent numbers in this series converge
to the golden mean τ a positive root of the golden equation x2x1=0. Kap
praff names the classical golden mean τ the 1st Silver Mean of Type 1 and
denotes it by SM1(1).
For the case N=2, the recursive relation (4.238) at the seeds G(1)=1
and G(2)=2 generates Pell’s Sequence 1,2,5,12,29,... . The ratios of the ad
jacent numbers in Pell’s sequence converge to the positive root of the
equation x 22x1=0 Kappraff names the positive root of this equation
θ = 1 + 2 = 2.414... the 2nd Silver Mean of Type 1 and denotes it by SM1(2).
x−
4.12. Gazale Formulas
4.12.1. The Generalized Fibonacci mNumbers
Independently of Spinadel and Kappraff, the idea of generalized Fibonac
ci numbers was developed in the book [45] written by Egyptian mathemati
cian and engineer Midchat Gazale. Gazale considers the recursive relation
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
233
Fm(n+2)=m Fm(n+1)+Fm(n),
(4.240)
where m is a positive real number and n=0,±1,±2,±3,… . Note that the recur
sive formula (4.240) is similar to the recursive formula (4.206) used by Spi
nadel, when we take p=m and q=1 in (4.206). However, in contrast to (4.206),
where the coefficients p and q are integers, the coefficient m in the recursive
formula (4.240) used by Gazale is a positive real number m>0.
We will name a positive real number m, used in the recursive relation
(4.240), an order of the recursive relation (4.240). If we take the seeds
(4.241)
Fm(0)=0, Fm(1)=1
and then use the recursive relation (4.240) for a given m>0, we obtain an
infinite number of the recursive numerical sequences called Generalized Fi
bonacci Numbers of Order m or simply Fibonacci mnumbers.
Note that for the case m=1 the recursive relation (4.240) and the seeds
(2.41) can be represented, respectively, as follows:
F1(n+2)=F1(n+1)+F1(n)
(4.242)
F1(0)=0, F1(1)=1.
(4.243)
It is clear that the recursive relation (4.242) with the seeds (4.243) gener
ates the classical Fibonacci numbers.
For the case of m=2 the recursive formula (4.240) and the seeds (4.241)
are reduced to the following:
F2(n+2)=2F2(n+1)+F2(n)
(4.244)
(4.245)
F2(0)=0, F2(1)=1.
This case generates the socalled Pell numbers: 0, 1, 2, 5, 12, 29, … .
If we take m = 2, then the recursive relation (4.240) at the seeds (4.241)
generates the following recursive numerical sequence:
0, 1, 2 , 3, 4 2 , 11, 15 2 , 41, 56 2 ,....
4.12.2. The Generalized Golden Mean of Order m
Let us represent the recursive relation (4.240) as follows:
Fm (n + 2)
1
=m+
.
Fm (n + 1)
Fm (n + 1)
(4.246)
Fm (n)
For the case n→∞, the expression (4.246) is reduced to the following qua
dratic equation:
(4.247)
x2mx1=0.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
234
Equation (4.247) has two roots a positive root
4 + m2 + m
2
and a negative root
x1 =
(4.248)
− 4 + m2 + m
.
2
If we sum up (4.248) and (4.249), we obtain:
x2 =
(4.249)
(4.250)
x1+ x2=m.
If we substitute the root (4.248) for x in Eq. (4.247), we obtain the follow
ing identity:
(4.251)
x12 = mx1 + 1.
If we multiply or divide repeatedly all terms of the identity (4.251) by x1,
we obtain the following general identity:
(4.252)
x1n = mx1n −1 + x1n −2 ,
where n=0,±1,±2,±3,… .
Using similar reasoning for the root x2, we obtain the following identity
for the root x2:
(4.253)
x2n = mx2n −1 + x2n −2 .
Denote a positive root x1 by Fm and name it the Golden Mean of Order m
or the Golden mProportion. The golden mproportion Fm has the following
analytical expression:
4 + m2 + m
(4.254)
.
2
Note that for the case m=1, the golden mproportion coincides with the
classical golden mean Φ1 = 1 + 5 2 .
Let us express the root x2 by the golden mproportion Φm. After simple
transformation of (4.249) we can write the root x2 as follows:
Φm =
(
x2 =
)
− 4 + m2 + m
−4
1
=
=−
.
2
2
Φm
2( 4 + m + m)
(4.255)
Substituting Fm for x1 and (1/Fm) for x2 in (4.250), we obtain:
m = Φm − 1 / Φm ,
where Φm is given by (4.254) and 1/Φm is given by the formula:
(4.256)
1
4 + m2 − m
=
.
2
Φm
(4.257)
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
235
Using the formulas (4.254) and (4.257), we can write the following identity:
1
Φm +
= 4 + m2 .
(4.258)
Φm
It is also easy to prove the following identity:
Φmn = mΦmn −1 + Φmn −2 ,
where n=0, ±1, ±2, ±3,… .
(4.259)
4.12.3. Two Surprising Representations of the Golden mProportion
For the case n=2, the identity (4.259) can be represented in the form:
(4.260)
Φm2 = 1 + mΦm .
The following representation of the golden mproportion Φm comes from
(4.260):
Φm = 1 + mΦm .
(4.261)
Substituting 1+ mΦm for Φm on the righthand part of (4.261), we obtain:
Φm = 1 + m 1 + Φm .
(4.262)
Continuing this process ad infinitum, that is, substituting repeatedly
1+ mΦm for Φm on the righthand part of (4.262), we obtain the following sur
prising representation of Φm:
Φm = 1 + m 1 + m 1 + m ... .
Now, represent the identity (4.260) in the form:
1
Φm = m +
.
Φm
(4.263)
(4.264)
Substituting m+(1/Φm) for Φm on the righthand part of (4.264), we obtain:
1
Φm = m +
.
1
(4.265)
m+
Φm
Continuing this process ad infinitum we obtain the following surprising
representation of the golden mproportion:
1
Φm = m +
.
1
m+
(4.266)
1
m+
m + ...
Alexey Stakhov
THE MATHEMATICS OF HARMONY
236
Note that for the case of m=1, the representations (4.263) and (4.28) coin
cide with the wellknown representations of the classical golden mean in the
forms (1.20) and (1.24), respectively.
4.12.4. A Derivation of the Gazale Formula
The formula (4.240) at the seeds (4.241) defines the Fibonacci mnumbers
Fm(n) by recursion. We can represent the numbers Fm(n) in analytical form us
ing the golden mproportion Φm.
Let us represent the Fibonacci number Fm(n) by the roots x1 and x2 in the
form:
(4.267)
Fm (n) = k1 x1n + k2 x2n ,
where k1 and k2 are constant coefficients that are the solutions to the follow
ing system of algebraic equations:
 Fm (0) = k1 x10 + k2 x20 = k1 + k2
.

(4.268)
1
1
 Fm (1) = k1 x1 + k2 x2 = k1Φm − k2 (1 / Φm )
Taking into consideration that Fm(0)=0 and Fm(1)=1, we can rewrite the
system (4.268) as follows:
k1 = k2
(4.269)
k1Φm + k1 (1 / Φm ) = k1 ( Φm + 1 / Φm ) = 1.
(4.270)
Taking into consideration (4.269) and (4.270) and also the identity (4.258),
we can find the following formulas for the coefficients k1 and k2:
1
1
; k2 = −
.
k1 =
(4.271)
4 + m2
4 + m2
Taking into consideration (4.271), we can write the formula (4.267) as
follows:
1
1
1
Fm (n) =
x1n −
x2n =
x1n − x2n .
(4.272)
2
2
2
4+m
4+m
4+m
Taking into consideration that x1=Φm and x2=1/Φm, we can rewrite the
formula (4.272) as follows:
Φ n − (−1 / Φm )n
Fm (n) = m
(4.273)
4 + m2
or
(
)
n
n

2 
 m − 4 + m2  
m
+
+
m
4

 −
 .
Fm (n) =
2 


 
2
2
4+m 
 
 

1
(4.274)
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
237
For the partial case m=1, formula (4.274) is reduced to the formula:
n
n
1  1 + 5   1 − 5  

F1 (n) =
 
 −
(4.275)
5  2   2  


called the Binet formula. This formula was obtained by Binet in 1843, although
the result was known to Euler, Daniel Bernoulli, and de Moivre more than
one century earlier. In particular, de Moivre obtained this formula in 1718.
For the case m=2, the formula (4.274) takes the following form:
n
n
1 
F2 (n) =
1+ 2 − 1− 2 .
(4.276)

2 2 
(
) (
)
Note that this formula was obtained for the first time by the English math
ematician John Pell (16101685).
For the cases m=3 and m = 2 , the formula (4.274) takes the following
forms, respectively:
n
n
1  3 + 13   3 − 13  

F3 (n) =
 
 −
(4.277)
2  
2  
13 


n
n
1  2 + 6   2 − 6  

F 2 (n) =
  .
 − 
(4.278)
2
2
6 
 



Thus, the Egyptian mathematician Midhat J. Gazale obtained [45] the
unique mathematical formula (4.274), which includes as partial cases the Bi
net formula for Fibonacci numbers (4.275) for the case m=1 and Pell’s formu
la (4.276) for the case m=2. This formula generates an infinite number of the
Fibonacci mnumbers because every positive real number m generates its own
Generalized Binet Formula (4.274). Taking into consideration the uniqueness
of the formulas (4.272)(4.274) we will name this formula the Gazale Formula
for the Fibonacci mnumbers.
4.13. Fibonacci and Lucas mNumbers
4.13.1. Fibonacci mNumbers
Let us prove that Gazale formulas (4.272)(4.274) really express all Fi
bonacci mnumbers given by the recursive formula (4.240) at the seeds (4.241).
In fact, for the case n=0 the formula (4.274) gives the Fibonacci mnumber
Alexey Stakhov
THE MATHEMATICS OF HARMONY
238
Fm(0)=0 that corresponds to the seeds (4.241). For the case n=1, we can rewrite
the formula (2.274) as follows:
 m + 4 + m2 m − 4 + m2 

 =1
−

2
2
4 + m2 

1
Fm (1) =
that corresponds to the seeds (4.241).
This means that the formula (4.276) does express the seeds (4.241).
Suppose that the formula (4.274) is valid for a given n (the inductive hy
pothesis) and prove that this formula is valid for the case n+1, that is,
1
Fm (n + 1) =
x1n +1 − x2n +1 .
(4.279)
2
4+m
Using the identities (4.252) and (4.253), we can represent the formula
(4.279) as follows:
m
1
Fm (n + 1) =
x1n − x2n ) +
(
( x1n−1 − x2n−1 )
2
2
4+m
4+m
(4.280)
= mFm ( n ) + Fm ( n − 1) .
(
)
Thus, the formula (4.280), in fact, defines the Fibonacci mnumbers given
by the recursive relation (4.240) at the seeds (4.241).
Note that the formula (4.274) defines all Fibonacci mnumbers in the range
n=0, ±1, ±2, ±3, … . Let us find some surprising properties of the Fibonacci m
numbers. First of all, compare Fm(n) and Fm(n). We can write the formula
(4.273) as follows:
Fm (n) =
Φmn − (−1)n Φm− n
.
(4.281)
4 + m2
Let us consider the formula (4.281) for the negative values of n, that is,
Fm (− n) =
Φm− n − (−1)− n Φmn
.
(4.282)
4 + m2
Comparing the expression (4.281) and (4.282) for the even (n=2k) and
odd (n=2k+1) values of n, we find:
(4.283)
Fm(2k)= Fm(2k) and Fm(2k+1)=Fm(2k1).
This means that the sequences of the Fibonacci mnumbers in the range
n=0, ±1, ±2, ±3, … are symmetrical sequences with respect to the Fibonacci
mnumber Fm(0)=0 except that the Fibonacci mnumbers Fm(2k) and Fm(2k)
are opposite in sign.
In Table 4.5 we can see the Fibonacci mnumbers for the cases m=1, 2, 3, 4.
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
239
Table 4.5. Fibonacci mnumbers (m=1, 2, 3, 4)
m
4
3
2
1
0
1
2
3
4
1
3
2
1
1
0
1
1
2
3
2
12
5
2
1
0
1
2
5
12
3
33
10
3
1
0
1
3
10
33
4
72
17
4
1
0
1
4
17
72
4.13.2. The Generalized Cassini Formula
Next let’s find a fundamental formula that connects three adjacent Fi
bonacci mnumbers. For the case m=1, this formula is known as the Cassini
Formula. Remember that this formula for the classical Fibonacci numbers F1(n)
has the following form:
F12 ( n ) − F1 ( n − 1) × F1 ( n + 1) = ( −1)
n +1
.
(4.284)
It is easy to prove the following general identity for the Fibonacci mnumbers:
n +1
Fm2 ( n ) − Fm ( n − 1) × Fm ( n + 1) = ( −1) .
(4.285)
For example, for the case m=2 the Fibonacci mnumbers F2(5)=29,
F2(4)=12 and F2(3)=5 are connected by the following correlation: (12)2
(29×5)=1 and for the case m=3 the Fibonacci mnumbers F3(4)=33, F2(3)=10
and F3(3)=3 are connected by the following correlation: (10)2(33×3)=1. Note
that both examples correspond to the general formula (4.285).
4.13.3. Lucas mNumbers
Once again, consider the formula (4.267) that defines the Fibonacci mnum
bers. By analogy with the classical Lucas numbers we can consider the formula
(4.286)
Lm ( n ) = x1n + x2n .
It is clear that for the case m=1, this formula defines the classical Lucas
numbers: 2, 1, 3, 4, 7, 11, 18, … . Let us assume that this formula defines the
Generalized Lucas Numbers of Order m or simply Lucas mNumbers. For a giv
en m we can find some peculiarities of the Lucas mnumbers. First of all, cal
culate the seeds of the Lucas mnumbers given by (4.286). In fact, for the
cases n=0 and n=1 we have from (4.286), respectively:
Lm ( 0 ) = x10 + x20 = 1 + 1 = 2
(4.287)
Lm (1) = x11 + x21 = Φm + ( −1 / Φm ) = m.
(4.288)
THE MATHEMATICS OF HARMONY
Alexey Stakhov
240
Note that for the case m=1, the seeds (4.287) and (4.288) come for the seeds
of the classical Lucas numbers: L1(0)=2, L1(1)=1.
Using (4.252) and (4.253), we can represent the formula (4.286) as follows:
Lm ( n ) = x1n + x2n = mx1n −1 + x1n −1 + mx2n −1 + x2n −1
(4.289)
= m x1n −1 + x2n −1 + x1n −1 + x2n −1 .
(
) (
(
)
) (
)
Taking into consideration the definition (4.286), we can rewrite (4.289)
in the following form of a recursive relation:
(4.290)
Lm(n)=mLm(n1)+Lm(n2).
It is clear that the recursive relation (4.290) at the seeds (4.287) and
(4.288) gives the Lucas mnumbers in recursive form.
If we substitute x1=Φm and x2=1/Φm in the formula (4.286), we can repre
sent the Lucas mnumbers in the following analytical form:
n
Lm ( n ) = Φmn + ( −1 / Φm ) .
(4.291)
Although this formula is absent in Gazale’s book [45], we will name this
important formula the Gazale Formula for Lucas mNumbers.
We can rewrite the formula (4.291) as follows:
n
Lm ( n ) = Φmn + ( −1) Φm− n .
(4.292)
Let us write the formula (4.292) for the negative values of n, that is,
n
Lm ( − n ) = Φm− n + ( −1) Φmn .
(4.293)
Comparing the expressions (4.292) and (4.293) for even (n=2k) and odd
(n=2k+1) values of n, we obtain:
(4.294)
Lm(2k)=Lm(2k) and Lm(2k+1)=Lm(2k1).
This means that the sequences of Lucas mnumbers in the range n=0, ±1,
±2, ±3,… are symmetrical sequences with respect to the Lucas mnumber
Lm(0)=2 except that the Lucas mnumbers Lm(2k+1) and Lm(2k1) are oppo
site by sign.
In Table 4.6 we can see the Lucas mnumbers for the cases m=1, 2, 3, 4.
Table 4.6. The Lucas mnumbers (m=1, 2, 3, 4)
m
4
3
2
1
0
1
2
3
4
1
7
4
3
1
2
1
3
4
7
2
34
14
6
2
2
2
6
14
34
3
119
36
11
3
2
3
11
36
119
4
322
76
18
4
2
4
18
76
322
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
241
Note that for the case m=1 the Lucas mnumbers coincide with the classical
Lucas numbers and for the case m=2 coincides with the PellLucas numbers.
Note similar results obtained by Kappraff and Adamson in [153].
4.14. On the mExtension of the Fibonacci and Lucas pNumbers
4.14.1. A Recursive Relation for the Fibonacci (p,m)Numbers
Now, we define the recursive relation for the mextension of the Fibonac
ci pnumbers [154]. For a given integer p>0 and positive real number m>0
the recursive relation is given as follows
Fp,m(n)=mFp,m(n1)+Fp,m(np1)
with initial conditions
(4.295)
Fp,m(0)=a0, Fp,m(1)=a1, Fp,m(2)=a2, …, Fp,m(p)=ap,
where a0, a1, a2, … , ap are integer, real or complex numbers.
In particular, we can take these initial conditions as follows
(4.296)
Fp,m(0)=0, Fp,m(k)=mk1,
where k=1,2,3,…,p.
We name a new class of recursive numerical sequences given by (4.295) at
the seeds (4.296) an mextension of the Fibonacci pnumbers or simply Fibonacci
(p, m)numbers.
It is clear that the recursive formula (4.295) at the seeds (4.296) defines a
more general class of recursive numerical sequences than the Fibonacci pnum
bers or the Fibonacci mnumbers. Note that for the case m=1 the Fibonacci
(p,m)numbers coincide with the Fibonacci pnumbers, that is, Fp,1(n)= Fp (n),
and for the case p=1, the Fibonacci (p,m)numbers coincide with the Fibonacci
mnumbers, that is, F1,m(n)=Fm (n). For the cases p=1 and m=1, the Fibonacci
(p, m)numbers coincide with the classical Fibonacci numbers.
4.14.2. Some Properties of the Fibonacci (p,m)Numbers
Let us consider the mextension of the Fibonacci pnumbers given by
(4.295) with the initial conditions of (4.296). Let us calculate the set of Fi
bonacci (p,m)numbers for the negative values of the argument m
Fp,m(1), Fp,m(2), … , Fp,m(p), … , Fp,m(2p+1).
THE MATHEMATICS OF HARMONY
Alexey Stakhov
242
According to (4.296), we have: F p,m(p+1)=mp and Fp,m(p)=mp1, thus
Fp,m(0)=0. Continuing in this way, we obtain
(4.297)
Fp,m(1)=Fp,m(2)=…=Fp,m(p+1)=0.
Let us write the mextension of the Fibonacci pnumber Fp,m(1) in the form
Fp,m(1)=mFp,m(0)+Fp,m(p).
Then, we obtain
Fp,m(p)=1.
Using (4.295), (4.296) and (4.298), we have
(4.298)
Fp,m(p1)= Fp,m(p2)=…= Fp,m(2p+1)= 0.
Also, we have
(4.299)
(4.300)
Fp,m(2p)=m, Fp,m(2p1)=1, Fp,m(2p2)= 0,….
For the case m=2 we obtain the 2extension of Fibonacci pnumbers called
Pell pnumbers. Table 4.7 gives the values of Pell pnumbers for the cases of
p=1, 2, 3, 4.
Table 4.7. Pell pnumbers (m=2; p=1,2,3,4)
n
5
4
3
2
1
0
1
2
3
4
5
F1,2(n)
29
12
5
2
1
0
1
2
5
12
29
F2,2(n)
22
9
4
2
1
0
0
1
0
2
1
F3,2(n)
17
8
4
2
1
0
0
0
1
0
0
F4,2(n)
16
8
4
2
1
0
0
0
0
1
0
4.14.3. Characteristic Algebraic Equation for the Fibonacci (p, m)
Numbers
Let us represent the recursive relation (4.295) in the form:
Fp ,m (n)
Fp ,m (n −1)
= m+
=
1
Fp ,m (n −1)
mFp ,m (n −1) + Fp ,m (n − p −1)
= m+
Fp ,m (n − p −1)
Fp ,m (n −1)
1
Fp ,m (n −1) ⋅ Fp ,m (n − 2) ⋅⋅⋅ Fp ,m (n − p)
.
(4.301)
Fp ,m (n − 2) ⋅ Fp ,m (n − 3) " Fp ,m (n − p −1)
Suppose that
lim
n→∞ F
Fp,m (n)
p ,m (n − 1)
= x.
(4.302)
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
243
Taking into consideration the definition (4.302) and directing n → ∞, we
can replace the expression (4.301) with the following algebraic equation
1
x =m+ p ,
x
whence it appears
xp+1mxp1=0.
(4.303)
We will name the equation (4.303) a Characteristic Equation for the Fi
bonacci (p, m)numbers.
Note that for the case m=1, Eq. (4.303) is reduced to Eq. (4.42) – the
characteristic equation for the Fibonacci pnumbers – and for the case p=1 to
Eq. (4.247) the characteristic equation for the Fibonacci mnumbers.
Equation (4.303) has p+1 roots x1, x2, …, xp, xp+1. If we substitute the root
xk (k=1, 2, 3, …, p+1) for x into Eq. (4.303), we obtain the following identity
for the root xk:
(4.304)
x kp +1 = mx kp + 1.
If we multiply and divide all terms of the identity (4.304) by xk repeated
ly, we come to the following general identity
(4.305)
x kn = mx kn −1 + x kn − p −1 = x k × x kn −1 ,
where n=0,±1,±2,±3,… and xk (k=1,2,3,…, p+1) is a root of Eq. (4.303).
4.14.4. The Golden (p,m)Proportions
According to Descartes’ “rule of signs,” Eq. (4.303) has the only positive
root. Suppose without loss of generality that the root x1 is a positive root of
Eq. (4.303). Let us denote the root x1 by Φp,m and name it a Golden (p, m)
Proportion. Substituting Φp,m for xk (k=1) into the identities (4.304) and
(4.305), we obtain two important identities for Φp,m:
Φpp,+m1 = mΦpp,m + 1
(4.306)
Φpn,m = Φpn,−m1 + mΦpn,−mp −1 = Φp,m × Φpn,−m1 ,
(4.307)
where n=0,±1,±2,±3,….
If we divide all terms of the identity (4.306) by Φpp,m , we obtain the fol
lowing remarkable property of the golden (p,m)proportions
Φp,m = m + 1 / Φpp,m
or
(4.308)
Φp,m − m = 1 / Φpp,m .
(4.309)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
244
Note that the golden (p,m)proportions Φp,m is a wide generalization of the
golden pproportions (m=1), the generalized golden mproportions (p=1) and
the classical golden mean (p=1, m=1). Also the identities (4.308) and (4.309)
are a wide generalization of the corresponding identities for the golden ppro
portions, the golden mproportions, and the classical golden mean.
4.14.5. Properties of the Roots of the Characteristic Equation
Theorem 4.4. For a given integer p>0 and a positive real number m, the
following correlations for the roots of the golden algebraic equation
xp+1mxp1=0 are valid:
x1 + x 2 + x 3 + ... + x p + x p+1 = m
( x1 x2 + ... + x1 x p+1 ) + ( x2 x3 + ... + x2 x p+1 )+ ... + ( x p−1 x p + x p−1 x p+1 ) + x p x p+1 = 0
( x x x + ... + x x x ) + ( x x x + ... + x x x ) + ... + x
1
2
3
1
p
p +1
2
3
4
2
p
p+1
p−1
(4. 310)
x p x p+1 = 0
...
x1 x2 " x p−2 x p−1 x p + x1 x 3 x 4 " x p−1 x p x p+1 + ... + x2 x 3 x 4 "x p−1 x p x p+1 = 0
(4.311)
x1 x2 x 3 " x p−1 x p x p+1 = (−1) .
p
(4.312)
Theorem 4.4 is proved by analogy with Theorem 4.1. We use the “Basic
Theorem of Algebra” that is given by (4.74) to represent the characteristic
equation (4.303) in the form:
x p+1 − mx p −1 = ( x − x1 )( x − x2 )"( x − x p+1 ) = x p+1 −( x1 + x 2 + ... + x p+1 ) x p
+( x1 x2 + ... + x1 x p+1 + x2 x3 + ... + x2 x p+1 + ... + x p−1 x p+1 + x p x p+1 ) x p−1
−( x1 x2 x3 + ... + x1 x p x p+1 + x2 x3 x4 + ... + x2 x p x p+1 + ... + x p−1 x p x p+1 ) x p−2
+( x1 x2 x 3 x 4 + x1 x 2 x 3 x 5 + ... + x p−2 x p−1 x p x p+1 ) x p−3
(4.313)
...
x p−2 x p−1 x p + ... + x2 x 3 x 4 " x p−1 x p x p+1 ) x − x1 x2 x 3 " x p−1 x p x p+1 = 0.
+( x1 x2 x 3 "x
We can give some explanations regarding the identities (4.310), (4.311),
and (4.312) that connect the roots of Eq. (4.303). It is evident from (4.310)
that the sum of the roots of Eq. (4.303) is identically equal to m. The expres
sion (4.311) gives the values for every possible sum of the roots of Eq. (4.303)
taken by two, three, ..., or р roots from the (р+1) roots of Eq. (4.303). Accord
ing to (4.311), each of these sums is equal to 0 ! At last, the expression (4.312)
gives the value of the product of all roots of Eq. (4.303). According to (4.312)
this product is equal to 1 (for the even р) or 1 (for the odd р).
Theorem 4.5. For a given integer p=0,1,2,3,… and for the condition, when
k takes its values from the set {1,2,3,…, p}, the following identity is valid for
the roots of the algebraic equation xp+1mxp1=0:
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
245
( x + x + x + ... + x + x ) = x + x + x + ... + x + x
k
k
1
k
2
k
3
k
p
k
k
p +1 = m .
(4.314)
Theorem 4.5 is proven by analogy with Theorem 4.2. For the proof we
may consider the following expression:
1
2
3
p
p +1
( x + x + x + ... + x + x ) ,
k
1
2
3
p
(4.315)
p +1
where k takes its values from the set {1,2,3,…, p}.
Taking into consideration the identity (4.310), we can write:
( x + x + x + ... + x + x ) = m .
k
k
(4.316)
On the other hand, if we factorize the expression (4.315) and take into
consideration the identities (4.311), we obtain:
k
( x1 + x2 + x3 + ... + x p + x p +1 ) = x1k + x2k + x3k + ... + x pk+1 ,
(4.317)
1
2
3
p
p +1
whence the identity (4.314) appears.
4.14.6. Generalized Binet Formulas for the Fibonacci (p,m)Numbers
For a given p > 0, the generalized Binet formula for the Fibonacci (p,m)
numbers is
n
n
n
Fp,m ( n ) = k1 ( x1 ) + k2 ( x2 ) + ... + kp +1 ( x p +1 ) ,
(4.318)
where x1, x2, …, xp+1 are the roots of the characteristic equation (4.303) and k1,
k2, …, kp+1 are constant coefficients that depend on the initial terms (4.296) of
the Fibonacci (p,m)sequence.
In order to calculate the values of the coefficients k1, k2, …, kp+1 in (4.318),
we consider solutions of the following system of equations
 Fp,m (0) = k1 + k2 + " + kp +1 = 0

 Fp,m (1) = k1 x1 + k2 x2 + " + kp +1 x p +1 = 1

2
2
2
.
 Fp,m (2) = k1 x1 + k2 x2 + " + kp +1 x p +1 = m

 #
 Fp, m ( p) = k1 x1p + k2 x2p + " + kp +1 x pp+1 = m p −1

(4.319)
Solving the system of the equations (4.319), we can obtain the numerical
values of the coefficients k1, k2, …, kp+1 for different values of p.
For the case p=1, the Fibonacci (p, m)numbers are reduced to the Fi
bonacci mnumbers, and we obtain the formula similar to the Gazale formula
(4.281), that is,
F1,m (n) =
Φ1n,m − (−1)n Φ1−,mn
4 + m2
,
(4.320)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
246
where Φ1,m is the golden (1, m)proportion.
For the case p=2, the recursive relation (4.295), the seeds (4.296), and
the characteristic equation (4.303) take the following forms, respectively,
F2,m(n)=mF2,m(n1)+F2,m(n3)
(4.321)
F2,m(0)=0, F2,m(1)=1, F2,m(2)=m
(4.322)
(4.323)
x3mx21=0.
Equation (4.323) has one real root and two complex roots. The roots of
Eq. (4.323) are:
h2 + 2mh + 4m2
,
6h
 h m2 
h2 − 2mh + 4m2
+i 3 −
x2 = −
,
6h
 12 3h 
 h m2 
h2 − 2mh + 4m2
−i 3 −
x3 = −
,
6h
 12 3h 
x1 =
(4.324)
(4.325)
(4.326)
where h = 3 108 + 8m 3 + 12 81 + 12m 3 .
The Binet formula for the Fibonacci (2,m)numbers is
F2,m ( n ) = k1 ( x1 ) + k2 ( x2 ) + k3 ( x3 ) .
n
n
n
(4.327)
The values of k1, k2, k3 are solutions of the system
F2,m ( 0 ) = k1 + k2 + k3 = 0,
F2,m (1) = k1 x1 + k2 x2 + k3 x 3 = 1,
F2,m ( 2) = k1 ( x1 ) + k2 ( x2 ) + k3 ( x 3 ) = m.
2
2
2
(4.328)
Solving the system, we obtain
k1 =
2h(h + 2m)
h[−(h + 2m) − i 3 (h − 2m)]
3
3 , k2 =
h + 8m
h 3 + 8m 3
k3 =
h[−(h + 2m) + i 3 (h − 2m)]
.
h 3 + 8m 3
Taking p=2,3,4,5,… we can derive the generalized Binet formulas for the
Fibonacci (p,m)numbers. For example, the Binet formula for the Fibonacci
(2, m)numbers is the following:
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
247
2h(h + 2m)  h2 + 2mh + 4m2 
F2,m (n) = 3

h + 8m 3 
6h

n
h[−(h + 2m) − i 3 (h − 2m)]  h2 + 4m2 − 4hm
 h m2  
+
i
−
+
3

 12 − 3h  
h 3 + 8m 3
12h



n
h[−(h + 2m) + i 3 (h − 2m)]  h + 4m − 4hm
 h m 
−i 3 −
−

3
3
12h
h + 8m
 12 3h  

n
+
2
2
2
(4.329)
4.14.7. Generalized Binet Formulas for the Lucas (p,m)Numbers
By analogy to the formula (4.161), we introduce the following formula
that defines the Lucas ( p, m ) numbers:
Lp,m ( n ) = ( x1 ) + ( x2 ) + ... + ( x p +1 ) ,
n
n
n
(4.330)
where x1, x2, …, xp+1 are the roots of the characteristic equation (4.303).
Calculate the value of the initial (p+1) terms of the number sequence de
fined by (4.330). For the case n=0, we have:
Lp,m ( 0 ) = ( x1 ) + ( x2 ) + ... + ( x p +1 ) = p + 1.
0
0
0
(4.331)
Let us take k from the set {1, 2, 3,..., p} and represent the formula (4.330)
in the form:
Lp,m ( k ) = ( x1 ) + ( x2 ) + ... + ( x p +1 ) .
Using the identity (4.314), we can write:
k
k
k
(4.332)
Lp,m(k)=mk,
(4.333)
where k takes its values from the set {1,2,3,…, p}.
The expressions (4.331) and (4.333) can be considered as the seeds of the
Lucas (p, m)numbers that are given by the following recursive formula:
(4.334)
Lp,m(n)=mLp,m(n1)+Lp,m(np1).
Let us prove that the formula (4.330) gives the same numerical sequence
as well as the recursive relation (4.334) at the seeds (4.331) and (4.333).
Suppose that the formula (4.330) defines the same number Lp,m(n) that is
calculated according to the recursive relation (4.334). Consider the formula
(4.330) for the case of n+1, that is,
Lp,m ( n + 1) = ( x1 )
n +1
+ ( x2 )
n +1
+ ... + ( x p +1 )
n +1
.
(4.335)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
248
Using the identity (4.305), we can write the expression (4.335) as follows
n
n
n
Lp,m ( n + 1) = m  ( x1 ) + ( x2 ) + ... + x p +1 


(4.336)
−
−
1
n
p
n − p −1
n − p −1
+ ( x1 )
+ ( x2 )
+ ... + x p+1
,
(
(
)
)
whence it appears
Lp,m(n+1)=mLp,m(n)+Lp,m(np)
(4.337)
that corresponds to the recursive relation (4.334).
This means that our reasoning results in the discovery of a new class of
numerical sequences – the mextension of the Lucas pnumbers or simply the
Lucas (p, m)numbers.
Note that for the case m=1 the recursive relation (4.334) and the seeds
(4.331) and (4.333) are reduced to the following:
Lp,1(n)=Lp,1(n1)+Lp,1(np1)
(4.338)
(4.339)
Lp,1(0)=p+1 and Lp,1(k)=1 (k=1,2,3,…,p).
It is clear that the recursive relation (4.338) at the seeds (4.339) gives the
above Lucas pnumbers.
For the case m=2, the recursive relation (4.334) and the seeds (4.331) and
(4.333) are reduced to the following:
Lp,2(n)=2Lp,2(n1)+Lp,2(np1)
(4.340)
Lp,2(0)=p+1; Lp,2(k)=2k (k=1,2,3,…,p).
(4.341)
For the case p=1, the Lucas (p,m)numbers are reduced to the Lucas m
numbers and we obtain the Binet formula similar to (4.292), that is,
(4.342)
L1,m ( n ) = Φ1n,m + ( −1) Φ1−,mn ,
where Φ1,m is the golden (1, m)proportion.
For the case p=2, the recursive relation (4.334), the seeds (4.331) and
(4.333), the characteristic equation (4.303), and Binet formula (4.330) take
the following forms, respectively:
n
L2,m(n)=mL2,m(n1)+L2,m(n3)
(4.343)
L2,m(0)=3, L2,m(1)=2, L2,m(2)=4
(4.344)
x3mx21=0
(4.345)
(4.346)
L2. m ( n ) = ( x1 ) + ( x2 ) + ( x3 ) .
If we substitute the expressions (4.324)(4.326) for the roots x1, x2, x3
into (4.346), we obtain the following analytical expression of the Binet for
mula for the Lucas (2,m)numbers that are called the PellLucas pnumbers:
n
n
n
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
249
n
 h 2m2 m   h m2 m i 3  h 2m2  
L2,m (n) =  +
+  + − −
+ +
−

3   12 3h 3
2  6 3h  
 6 3h
n
n
(4.347)
 h m 2 m i 3  h 2m 2  
+ − −
+ −
−
.

2  6 3h  
 12 3h 3
Table 4.8 gives the 2extension of the Lucas pnumbers (PellLucas p
numbers) for the cases p=1,2,3,4.
Table 4.8. PellLucas pnumbers (m=2, p=1,2,3,4)
n
5
4
3
2
1
0
1
2
3
4
5
L1,2(n)
82
34
14
6
2
2
2
6
14
34
82
L2,2(n)
10
8
3
4
0
3
2
4
11
24
52
L3,2(n)
0
4
6
0
0
4
2
4
8
20
42
L4,2(n)
5
8
0
0
0
5
2
4
8
16
37
It is clear that our reasoning resulted in a wide generalization of the Fibonacci
and Lucas pnumbers and the Fibonacci and Lucas mnumbers. The Fibonacci
and Lucas (p,m)numbers are of theoretical interest for discrete mathematics
and open new perspectives for the development of theoretical physics because
new recursive numerical sequences and new mathematical constants that follow
from this approach may be discovered in some physical processes.
4.15. Structural Harmony of Systems
4.15.1. Soroko’s Law of the Structural Harmony of Systems
Belorussian philosopher Eduardo Soroko was one of the first researchers
who used the Fibonacci pnumbers and golden pproportions for simulation of
the processes in selforganizing systems [25, 56]. Soroko’s main idea is to con
sider real systems from the dialectical point of view. As is well known, any nat
ural object can be represented as the dialectical unity of the two opposite sides
A and B. This dialectical connection may be expressed in the following form:
A+B=U (universum).
(4.348)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
250
The equality (4.348) is the most general expression of the socalled
Conservation Law. Here A and B are distinctions inside of the Unity, logically
disjoint classes of the Whole. There is one requirement that A and B need to
be measured by the same measure and members of the relation that underlies
the unity. Probability and improbability of events, mass and energy, the nu
cleus of an atom and its envelope, substance and field, anode and cathode,
animals and plants, spiritual and material beginnings in a value system, and
profit and cost are various examples of (4.348).
The identity (4.348) may be reduced to the following normalized form:
(4.349)
A + B = 1,
where A and B are the relative “weights” of the parts A and B that make up
some Unity.
The Law of Information Conservation is a partial case of (4.348):
I+H=logN,
(4.350)
where I is an information quantity and H is the entropy of the system having
N different states.
The normalized form of the law (4.350) is the following:
(4.351)
R + H = 1,
where R=1/logN is a relative redundancy, H = H log N is a relative entropy.
Let us consider the process of system selforganization. This one is re
duced to the passage of the system into some “harmonious” state called the
state of thermodynamic equilibrium. There is some correlation or proportion
between the sides A and B of the dialectical contradiction (4.348) for the state
of thermodynamic equilibrium. This correlation has a strictly regular charac
ter and is the cause of system stability. Soroko uses the Principle of Multiple
Relations to find a connecting law between A and B in the state of thermody
namic equilibrium. This principle is well known in chemistry as Dalton’s Law
and in crystallography as the Law of Rational Parameters.
Soroko proposes the hypothesis that the Principle of Multiple Relations is
a general principle of the Universe. That is why, in accordance with this prin
ciple there is the following connection between the components R and H in
the identity (4.351):
log R = ( s + 1) H
or
(4.352)
(4.353)
log H = ( s + 1) R.
The equalities of (4.352) and (4.353) may be represented in exponential form:
Chapter 4
( )
R= H
Generalizations of Fibonacci Numbers and the Golden Mean
251
s +1
(4.354)
(4.355)
H = R s +1 ,
where the number s is called the Range of Multiplicity and takes the following
values: 0,1,2,3,… .
Substituting the expressions (4.354) and (4.355) into the equality (4.351),
we obtain the following algebraic equations, respectively:
(H )
s +1
+ H −1 = 0
(4.356)
(4.357)
R s +1 + R − 1 = 0.
If we denote the variables H and R in the equations (4.356) and (4.357)
by y, we obtain the following algebraic equation:
ys+1+y1=0.
(4.358)
Let us introduce the new variable x=1/y and apply it to the equation
(4.358). Inserting the new variable into (4.358), we obtain the following alge
braic equation:
(4.359)
xs+1+xs1=0.
We can see that the equation (4.359) coincides with the algebraic equa
tion of the golden pproportion that is given by (4.42). The positive root of
the equation (4.358) is the reciprocal of the golden pproportion:
(4.360)
βs=1/τs,
where τs is a positive root of the equation (4.359).
In accordance with Soroko’s concept, the roots of the equation (4.358)
that are equivalent to equation (4.359), express the Law of the Structural Har
mony of Systems:
“The generalized golden proportions are invariants that allow natu
ral systems in the process of their selforganization to find harmonious
structure, a stationary regime for their existence, and structural and func
tional stability.”
What is peculiar about Soroko’s Law? Since Pythagoras, scientists
combined the concept of Harmony solely with the classical golden mean
τ = 1 + 5 2. Soroko’s Law asserts that the harmonious state that corre
sponds to the classical golden mean is not unique to a system. Soroko’s
Law asserts that there are an infinite number of harmonious states of the
system that correspond to the numbers τ s or to the inverse numbers β s=1/
τ s (s=1,2,3,…) that are positive roots of the general algebraic equations
(4.358) and (4.359).
(
)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
252
Table 4.9 gives the values of Soroko’s “structural invariants” for the initial
terms of s.
Table 4.9. Soroko’s numerical invariants
s
βs
1
2
3
4
5
6
7
0.618 0.682 0.724 0.755 0.778 0.796 0.812
4.15.2. Application of Soroko’s Law to Thermodynamic and Information
Systems
The thermodynamic or informational state of a system is expressed by en
tropy, which is the principal concept of thermodynamics and information theo
ry. The expression for the entropy of an information source with the alphabet
A={a1, a2, …, aN} has the following form:
N
H =−
∑ p log p ,
k
(4.361)
k
k =1
where p1, p2, …, pN are the probabilities of the letters a1, a2, …, aN, N is a number
of the letters.
It is well known that entropy (4.361) reaches its maximum value
(4.362)
Hmax=logN
for the case, when the probabilities of the letters are equal among themselves,
that is,
p1=p2= …= pN=1/N.
Using the concept of relative entropy
H = H log N ,
we can write the following evident equality:
(4.363)
N
H log N = H = −
∑ p log p .
k
k
(4.364)
k =1
In accordance with the Law of the Structural Harmony of Systems, any
system reaches its harmonious state when its relative entropy (4.363) satis
fies the equality (4.353). The following expression for the entropy of the
harmonious system follows from this consideration:
N
H =−
∑ p log p = β log N .
k
k
s
(4.365)
k =1
It is clear that for any given s, the problem of obtaining the optimal set of
values pi (i=1, 2, 3, …, N) corresponding to the optimal (“harmonious”) state
Chapter 4
Generalizations of Fibonacci Numbers and the Golden Mean
253
of the system, has many solutions. However, the expression (4.365) plays a role
in the function’s “aim” towards the solution of various scientific and technical
problems.
In his book, Structural Harmony of Systems [25], Soroko gave a number of
interesting examples from different fields of science to confirm the Law of the
Structural Harmony of Systems. For example, let us consider such a natural
object as “dry air” that is the basis for life on Earth. We can ask the question:
does the “dry air” have an optimal or harmonious structure? Soroko’s theory
gives a positive answer to this question. In fact, the chemical compound of the
“dry air” is the following: nitrogen – 78.084%; oxygen – 20.948%; argon –
0.934%; carbon dioxide – 0.031%; neon – 0.002%; and helium – 0.001%. If we
calculate the entropy of the “dry air” according to the formula of (4.361) and
then its relative entropy according to (4.363) taking into consideration that
logN=log 6, then the value of the relative entropy of the “dry air” will be equal
to 0.683. With a high degree of precision this value corresponds to the invari
ant β2=0.682. This means that in the process of selforganization the “dry air”
reaches its optimal, harmonious structure. This example is very typical and
demonstrates that “Soroko’s Law” can be used today for checking the bio
sphere state, in particular, the states of air and water.
It is clear thus the practical use of the Law of the Structural Harmony of
Systems provides a clear advantage to the solution of many technological, eco
nomical, ecological and other problems. In particular, it can help improve the
technology of structurallycomplicated products, including the monitoring
of the biosphere and so forth.
4.16. Conclusion
In recent years the generalization of Fibonacci and Lucas numbers and
the golden section has been an important trend in the development of Fi
bonacci number theory. One may cite various ways in which this generaliza
tion takes place. One way, for example, is a generalization based on Pascal’s
triangle. In the latter half of the 20th century several eminent mathemati
cians (including Martin Gardner [12], George Polya [17], Alfred Renyi [23]
and others) each independently discovered a connection of Fibonacci num
bers with Pascal’s triangle and binomial coefficients. This finding confirms a
fundamental connection of Fibonacci and golden mean based Harmony Math
ematics with combinatorial analysis suggesting a future line of development.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
254
In the early 1970s, in his doctoral dissertation [19], Alexey Stakhov intro
duced the socalled Fibonacci pnumbers (p=0,1,2,3,…) that follow from the
diagonal sums in Pascal’s triangle. By studying the Fibonacci pnumbers, Sta
khov generalized the golden section problem and introduced the golden p
proportions that are positive roots of the characteristic equation xp+1=xp+1.
Later by continuing this research, Stakhov formulated the Generalized Prin
ciple of the Golden Section [107] and generalized the Euclidean problem of the
“division in extreme and mean ratio” (the golden section) [20]. Later Stakhov
and Rozin generalized the Binet formulas and introduced the generalized Lucas
pnumbers [111].
Fibonacci numbers were further generalization through a consideration
of the generalized recursive relation G(n+1)=pG(n)+qG(n1) (p and q are
integers or real numbers) that coincides with the classical recursive Fibonac
ci relation for the case p=q=1. This became the source of many original dis
coveries made by Vera W. de Spinadel [42], Midhat Gazale [45] and Jay Kap
praff [47]. By using this approach, Vera W. de Spinadel introduced the Me
tallic Means [42]. By using the recursive relation Fm(n+1)=mFm(n)+Fm(n1)
and the characteristic equation x2mx1=0, where m is a positive real number,
Midhat Gazale further generalized the Binet formula for Fibonacci mnum
bers Fm(n). Gazale formulas allow one to represent all Fibonacci and Lucas m
numbers by the golden mproportion Fm that is a generalization of the classi
cal golden mean.
Recently, E. Gokcen, Naim Tuglu and Alexey Stakhov [154] introduced
the mextension of the Fibonacci and Lucas pnumbers, generating the new
class of Fibonacci and Lucas (p,m)numbers. This new class of characteristic
equations xp+1mxp1=0, for generalized Fibonacci and Lucas (p,m)numbers
and the golden mean, infinitely extend algebraic equations and mathematical
constants, which can be used in contemporary mathematics, theoretical phys
ics and computer science.
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
255
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
5.1. The Simplest Elementary Functions
5.1.1. Trigonometric Functions
In mathematics, an Elementary Function is built up from a finite number
of exponentials, logarithms, constants, variables, and roots of equations by
using the four elementary operations (addition, subtraction, multiplication
and division). Logarithms, Exponential Functions (including Hyperbolic Func
tions), Power Functions, and Trigonometric Functions are the best known
amongst them.
We begin to study the elementary functions starting with trigonometric
functions. Trigonometry and trigonometric functions Sine, Cosine, Tangent, Co
tangent, Secant, and Cosecant are well known to many of us from secondary
school. Trigonometry was developed in antiquity initially as a branch of astron
omy, as a computing tool for practical purposes. Application of trigonometry to
astronomy explains why spherical trigonometry arose first, before planar trigo
nometry. Ancient Greek astronomers successfully solved some problems of trig
onometry. However, these scientists (Ptolemy, Menelaus, etc.) had not studied
trigonometric functions (sine, cosine, etc.) but line segments and spans. The span
that connects together a double arch 2α played a role of a sine of the angle α.
Ptolemy had found a formula for the definition of the span as the sum and differ
ence of two arches, the span of the half arch, and so on.
Sine and cosine were first introduced by Indian scientists. In India, the
doctrine of trigonometric values was named Goniometry as it started to de
velop. The further development of the doctrine of trigonometric values con
tinued in the 9th15th centuries in the countries of the Middle and the Near
East. The Arabian mathematicians introduced all basic trigonometric func
tions derivative from sine and cosine: tangent, cotangent, secant and cose
cant. They had proved the basic relations amongst trigonometric functions.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
256
For the first time Arabian mathematicians began to develop trigonometry as
a part of mathematics independent from astronomy.
In the 13th15th centuries, the further development of trigonometry in Eu
rope continued following the translation of mathematical and astronomical works
of Arabian and Greek science into Latin. The German scientist Regiomontanus
(14361476) was the most famous European mathematician of this epoch in the
field of trigonometry. His trigonometric work Five Books about Triangles of All
Kinds was of great importance for the further development of trigonometry dur
ing the 16th17th centuries.
On the threshold of the 17th century, an analytical direction in trigonometry
started to develop. Before the 17th century, the direction regarding triangles and the
calculation of the elements of geometric figures was the main objective of trigonom
etry and the doctrine of trigonometric functions had developed along geometrical
lines. Whereas in the 17th19th centuries, trigonometry gradually became one of the
branches of mathematical analysis. Trigonometry finds wide application in mechan
ics, physics and engineering, especially in the study of oscillatory movements and
other periodic processes. The development of the doctrine of oscillatory movements,
of sound, light and electromagnetic waves became central to the basic contents of
trigonometry. It is well known from physics that the equation of harmonious fluctu
ation (for example, of a variable electric current) appears as:
y = A sin ( ωt + α ) .
Sinusoids are graphs of harmonious fluctuations; therefore, in physics and
engineering the harmonious fluctuations are often called Sinusoidal Fluctuations.
Newton and Euler developed an analytical approach to trigonometry, and
in the first half of the 19th century the French scientist Fourier proved that
any periodic movement can be presented (with any degree of accuracy) in the
form of the sum of simple harmonious fluctuations. Presently trigonometry
is no longer considered an independent part of mathematics. Its major sec
tion – the Doctrine about Trigonometric Function – is a part of the theory of
functions; and its other section –the Decision of Triangles – is considered to
be part of geometry (planar and spherical).
5.1.2. The Power Function
The Power Function plays an important role in mathematics. The Power con
cept originally meant the product of a finite number of equal coefficients (a pow
er with a natural parameter), that is,
an=a×a×...×a (n times).
(5.1)
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
257
This definition possesses the following mathematical properties:
( ab ) = a n × bn ; ( a / b ) = a n / bn ;
n
n
( am ) = amn ; am / an = am−n .
n
(5.2)
Over the centuries, this concept was repeatedly generalized and enriched
by its contents. The concept of the 2nd and 3rd powers a2 and a3 appeared in
connection with the definition of the area of a square and the volume of a
cube. Already the Babylonians had made use of tables of the squares and
cubes of numbers. The titles Square and Cube for the 2nd and 3rd powers
are of ancient Greek origin. The practice of solving more and more complex
algebraic problems, led to the necessity to generalize the power concept
and its extension by means of the introduction of the powers of zero, and
negative and fractional numbers. The powers of zero and also negative and
fractional parameters were defined so that the actions of the power with a
natural basis given by (5.2) were applied to them. The principle here ob
served in the generalization of mathematical concepts is referred to as “a
permanence principle.”
The power concept (5.1) depends on two parameters, the numbers of a and n.
Depending on what parameter is chosen as a variable, it is possible to give two gener
alizations of the power concept (5.1). The first generalization is the power function:
y=xσ,
(5.3)
where x is a variable and σ is a given real number.
Note that for the cases σ=0 and σ=1 according to (5.3) we have, respectively:
y=1 and y=x.
(5.4)
The graph of the function y=1 is a straight line parallel to the axis OX, and
the graph of the function y=x is a bisector of the 1st and 3d coordinate angles.
For the case σ=2 the graph of the function (5.3) is a Parabola y=x2. For the
case σ=3 the graph of the function (5.3) is a Cubic Parabola y=x3. This curve had
been used by French mathematician Gaspard Monge (17461818), the father of
descriptive geometry, for finding the real roots of the cubic equations.
A derivative of the power function (5.3) is equal to the product of the power
parameter α multiplied by the power function with the parameter (α1), that is,
d α
x = αx α −1 .
dy
The power function (5.3) is a basic mathematical tool that is used widely in
many other fields as well, including economics, biology, chemistry, physics, and
computer science, with such applications as compound interest, population
growth, chemical reactions, wave behavior, and public key cryptography.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
258
5.1.3. An Exponential Function
However, there is another generalization of the power concept (5.1). Let us
examine the socalled exponential function given by the following expression:
(5.5)
y=ax,
where x is a variable and a>0 is any real number.
The exponential function (5.5) has many applications in the study of natu
ral and social phenomena. It is known, for example, that the decay of radioactive
substance is described by the function (5.5). If we denote by t0 a period of the
halfdecay, that is, a time interval necessary to get half of the initial mass m0,
then the mass of the substance via t years can be expressed as follows:
d α
x = αx α −1 .
dy
The exponential functions (5.5) posses the following Exponential Laws:
a0 = 1; a1 = a; a x + y = a x a y ; a xy = ( a x ) ;
y
1 / a x = a − x ; a x b x = ( ab ) .
x
These correlations are valid for all positive real numbers a and b and all
values of the variables x and y. The expressions that involve fractions and
roots can often be simplified by using exponential functions because:
1 a = a −1 ;
n
( )
ab = n a
b
b
= an ,
where n is a natural number.
In mathematics, the following partial case of the exponential function is
widely used:
y = ex ,
(5.6)
where e=2.71828183 is the base of natural logarithms. The graph of the exponen
tial function (5.6) is represented in Fig. 5.1.
y
5
The exponential function is climbing
4
slowly with the negative values of x, it is
climbing quickly with the positive values
3
of x, and is equal to 1 when x is equal to 0.
2
As a function (5.6) is the function of real
variable x, the graph of (5.6) is positive for
1
1
2
all values of x and is increasing (by viewing 5 4 3 2 1
x
lefttoright). This graph never touches the
1
xaxis, although it approaches the xaxis;
thus, the xaxis is a horizontal asymptote
2
to the graph.
Figure 5.1. Exponential function
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
259
A derivative of the exponential function (5.6) coincides with the function
(5.6), that is,
d x
e = ex .
dx
Note that only the function (5.6) has this unique property.
For the exponential functions with other bases we have:
d x
a = (ln a)a x .
dx
5.1.4. Logarithms
The theory of logarithms is based on the following simple reasoning. Let us
examine the exponential function (5.5). First we represent the variable x in (5.5)
in the form of a special function of y as follows:
( )
x = log a y = log a a x .
We name this function a Logarithm with base a. Depending upon the choice
of the base a there are different kinds of logarithms. For the case a=2 we have
Binary Logarithms log2x=lgx, for the case a=10 we have Decimal Logarithms
log10x and finally, for the case a=e we have Natural Logarithms logex=lnx.
Logarithms are used widely in many areas of science and engineering
where the magnitudes vary over a large range. Logarithmic scales are used,
for example, in the decibel scale for the loudness of sound, the Richter scale of
earthquake magnitudes, and the astronomical scale of stellar brightness.
A derivative and an indefinite integral of logaz are given respectively by
d
1
z(ln z − 1)
log a z =
; log a dz =
+ c.
dz
z ln a
ln a
∫
5.2. Hyperbolic Functions
5.2.1. Definition of the Hyperbolic Functions
The function
e x − e− x
sh( x ) =
2
is called the Hyperbolic Sine and the function
(5.7)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
260
e x + e− x
2
is called the Hyperbolic Cosine.
ch( x ) =
y
y = ch(x)
(5.8)
y = sh(x)
There is a similarity between trigono
metric and hyperbolic functions. Like the
4
trigonometric sine and cosine that are the
3
coordinates of the points on a circle, the hy
2
perbolic sine and cosine are the coordinates
1
of the points on a hyperbola. The hyper
4 3 2 1
0
1
3
5
2
4
x
bolic functions are defined on all numeri
1
cal axes. The hyperbolic sine is an odd func
2
tion that is increasing on all numerical axes.
3
The hyperbolic cosine is an even function
4
that is decreasing on the interval (∞; 0)
and increasing on the interval (0; +∞). The
Figure 5.2. Graphs of the
point (0; 1) is the minimum of this func
hyperbolic sine and cosine
tion (see Fig. 5.2).
By analogy with the trigonometric functions we can define hyperbolic tan
gent and cotangent:
sh( x )
ch( x )
th( x ) =
, cth( x ) =
.
ch( x )
sh( x )
y
5
Analytical definitions (5.7) and (5.8) of the hyperbolic functions can be used
for obtaining some important identities of the hyperbolic trigonometry. It is
well known that there are Trigonometric Identities, for example, the Pythagore
an Theorem for the Trigonometric Functions:
cos2a+sin2a=1.
We can prove similar identities for the hyperbolic functions:
2
 e x + e− x   e x − e− x 
ch x − sh x = 
 −

2
2

 

−2 x
−2 x
2x
2x
e +2+e
e −2+e
=1
=
−
4
4
2
2
2
2
(5.10)
2
 e x + e− x   e x − e− x 
ch x + sh x = 
 +

2
2

 

e2 x + 2 + e −2 x e2 x − 2 + e −2 x e2 x + e −2 x
=
= sh2 x .
=
+
4
4
2
2
(5.9)
2
(5.11)
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
261
We can prove the following properties for derivatives and integrals:
( shx ) ′ = chx ; ( chx ) ′ = shx ; ( thx ) ′ = 1 / ch2 x
∫ sh( x )dx = ch( x ) + C
∫ ch( x )dx = sh( x ) + C
∫ th( x )dx = ln[ch( x )] + C
1
∫ ch ( x )dx = th( x ) + C
2
1
∫ sh ( x ) dx = −cth( x ) + C
2
5.2.2. History and Applications of the Hyperbolic Functions
5.2.2.1. Lambert and Riccati
Although Johann Heinrich Lambert (17281777), a French mathemati
cian, is often credited with introducing hyperbolic functions, it was actually
Vincenzo Riccati (17071775), an Italian mathematician, who did this in the
middle of the 18th century. He studied these functions and used them to
obtain solutions for cubic equations. Riccati found the standard Addition for
mulas, similar to trigonometric identities, for hyperbolic functions as well as
their derivatives. He revealed the relationship between the hyperbolic func
tions and the exponential function. For the first time Riccati used the sym
bols sh and ch for the hyperbolic sine and cosine.
5.2.2.2. NonEuclidean Geometry
Among the mathematical works of ancient science, The Elements by Euclid is
of special importance. This famous work contains the fundamentals of ancient
mathematics: elementary geometry, number theory, algebra, the general theory
of relations and calculation of areas and volumes. Euclid summarized Greek math
ematics and created a stable foundation for further mathematical progress. The
Elements by Euclid is constructed as a deductive mathematical system: at first the
definitions, the postulates and the axioms are given, then the theorems are formu
lated and their proofs are given. Among the Euclidean axioms, the 5th Euclidean
axiom, the Axiom about Parallels is the most famous.
During almost 2 millennia many mathematicians tried to deduce this axiom
as a theorem from other Euclidean axioms. The problem of the 5th Euclidean
Alexey Stakhov
THE MATHEMATICS OF HARMONY
262
axiom was brilliantly solved for the first time by a professor of Kazan University,
the Great Russian mathematician Nikolay Lobachevsky (17921856) who devel
oped a nonEuclidean geometry in 1826. Lobachevsky’s geometry is also named
Hyperbolic Geometry because it is based on the hyperbolic functions. Indepen
dently Lobachevsky, the young Hungarian mathematician Janosh Bolyai had
also developed a similar nonEuclidean geometry. The first published work on
nonEuclidean geometry, Lobachevsky’s article About the Geometry Beginnings,
was published in 1829 in The Kazan Bulletin. Three years later Janosh Bolyai’s
work on nonEuclidean geometry, called the Appendix, was published in Latin.
After Gauss’ death it was clear that he also had developed a geometry similar to
those of Lobachevsky and Bolyai.
Lobachevsky had formulated a new axiom about parallel lines that, in
contrast to the Euclidean 5th axiom, was formulated as follows:
“Through any point outside of a given straight line, it is possible to draw
at least two different ‘parallel’ straight lines that are mutually disjoint with
the given straight line.”
Having replaced the 5th Euclidean axiom with this new axiom,
Lobachevsky developed a nonEuclidean geometry that is as logically correct
as Euclidean geometry.
We will not stop to detail all features of Lobachevsky’s geometry, as its
study goes far outside the limits of “Elementary mathematics,” studied in
high school. It is important to emphasize that, by studying trigonometric
relations of his geometry, Lobachevsky used the above hyperbolic functions,
that is, Lobachevsky’s geometry is an important confirmation of the funda
mental character of hyperbolic functions in the development of new geomet
ric models of the Universe.
5.2.2.3. Minkowski’s Fourdimensional World
The 20th century became a new stage in the evolution of spatial ideas in all
scientific spheres. Physics stimulated the global process of change regarding
geometric spatial ideas. Until the creation of nonEuclidean geometry, it was
not necessary to prove the mutual relation of Newton’s mechanics and Euclid
ean geometry. This fact was considered to be obvious. However, at the end of
the 19th century and the beginning of the 20th century many new physical
facts and observations, which did not fit the classical spatial representations,
were gathered together. Maxwell’s research on electrodynamics, Michelson’s
experiments on the measurement of the speed of light and other scientific facts
resulted in the problem of the correspondence of Euclidean geometry to the
real physical world. Einstein’s theory of relativity resulted in the explanation of
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
263
new physical facts, which concern relations between physics and classical geome
try. The conclusion about the nonEuclidean character of real spatial geometry
became the main outcome of Einstein’s special theory of relativity.
The need for attracting new geometric ideas arose within 20th century phys
ics. In particular, Einstein’s special relativity theory (1905) was a cause for this
research. In this theory for the first time in the history of physics, a mutual rela
tion between space and time became the objects of physical research. Relativity
theory proves that the metrics of real space and time are not absolute: they de
pend on dynamical conditions, in which the spatial and temporal measurements
are carried out. Relativity theory for the first time showed that space and time
are continuum and that their properties are integrally interconnected.
In 1908, that is, in three years after the promulgation of Einstein’s special
theory of relativity, the German mathematician Herman Minkowski presented
the geometric substantiation of the special relativity theory [37]. Minkowski’s
idea is characterized by two essential peculiarities. In the first place, the geomet
ric spatialtemporal model that was offered by him was fourdimensional: in this
model the spatial and temporal coordinates are connected in the common coor
dinate system. The position of the material point in Minkowski’s model was de
termined by the point M (x, y, z, t) called the World Point. Secondly, the geo
metric connection between the spatial and temporal coordinates in Minkows
ki’s system had a nonEuclidean character, that is, the given model reflected cer
tain special properties of real spacetime that could not be simulated in the frame
work of the “traditional” Euclidean geometry.
Geometrically, a connection between the spatial (x) and temporal (t) co
ordinates in Minkowski’s model is given by the Hyperbolic Turn, the move
ment that is similar to the traditional turn of the Cartesian system in Euclid
ean geometry. Here, the coordinates x and t of any point can be transformed
according to the formulas:
x ′ = xchψ + tshψ, t ′ = xshψ + tchψ,
where ψ is the angle of the hyperbolic turn and ch and sh are the hyperbolic sine
and cosine, respectively.
The original geometric interpretation of the wellknown Lorentz transfor
mations follows from the given model:
x ′ = ( x + vt )
(
1− (v c) ;
2
)
t ′ = t + vx c 2 


1− (v c) ,
2
where v is the speed of the system, с is the speed of light.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
264
Note that in Minkowski’s geometry Lorentz transformations are rela
tions of the hyperbolic trigonometry expressed in physics terms. Minkows
ki’s geometry uncovers the hyperbolic nature of all mathematical formulas
of relativity theory.
5.2.2.4. Vernadsky’s Ideas
Up until the time of relativity theory the mathematical apparatus for
biological research practically did not exist. Nevertheless the response of
biologists to discoveries in geometric physics was practically instantaneous.
Russian scientist Vladimir Vernadsky was one of the first scientists to give
serious attention to the geometric problems of biology [37]. Vernadsky’s pecu
liar approach was a holistic vision of the “spacetime” problem. Vernadsky ex
panded on the concepts of the “spacetime” problem. He took into consideration
the biological specificity of its study and discussed the realization in Nature of
general geometric (spatialtemporal) laws. Vernadsky gave attention to the prob
lem of biological symmetry and considered this to be the key biological problem.
He came to the conclusion that the explanation of biological symmetry is con
nected to the nonEuclidean character of the spatial geometry of living sub
stance. He also assumed that the geometry of biological “spacetime” differs from the
geometry of “spacetime” relativity theory, known as the “Fourdimensional
Minkowski world” [37].
According to his opinion, the issue of how nonEuclidean geometry is
embodied in living nature is the key problem of biological research.
Vernadsky’s ideas became a leitmotiv for scientific research in subsequent
periods of development in biology resulting in the creation of a new direction,
mathematical biology.
5.3. Hyperbolic Fibonacci and Lucas Functions (StakhovTkachenko’s
Definition)
5.3.1. A Definition of Hyperbolic Fibonacci and Lucas Functions
Fibonacci and Lucas functions were introduced, for the first time, in 1993 by
Alexey Stakhov and Ivan Tkachenko [98]. If we compare the hyperbolic func
tions (5.7) and (5.8) with Binet formulas (2.66) and (2.67), we can see that Binet
formulas are similar to the hyperbolic functions in their mathematical structure.
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
265
For a strong definition of the hyperbolic Fibonacci and Lucas functions we
can rewrite the formulas (2.68) and (2.67) respectively, as follows:
(
F2 k = τ2 k − τ−2 k
(
F2 k +1 = τ
2 k +1
+τ
2 k +1
−τ
L2 k = τ + τ
−2 k
L2 k +1 = τ
2k
)
5
−( 2 k +1)
(5.12)
)
5
− ( 2 k +1)
(5.13)
(5.14)
,
(5.15)
where the discrete variable k takes its values from the set {0, ±1, ±2, ±3,...} .
Comparing the formulas (5.12)(5.15) to the hyperbolic functions (5.7) and
(5.8), we can see that formulas (5.12) and (5.14) correspond in their structure to
the hyperbolic sine (5.7) and the formulas (5.13) and (5.15) correspond to the
hyperbolic cosine (5.8). This simple analogy underlies the Hyperbolic Fibonacci
and Lucas Functions [98].
By substituting the discrete variable k in the formulas (5.12)(5.15) for
the continuous variable x that takes its values from the set of real numbers,
we obtain the following four new elementary functions:
Hyperbolic Fibonacci sine
sF ( x ) = ( τ2 x − τ −2 x ) 5 .
(5.16)
Hyperbolic Fibonacci cosine
− 2 x +1
cF ( x ) = τ2 x +1 + τ ( )
5.
(
)
(5.17)
Hyperbolic Lucas sine
sL( x ) = τ2 k +1 − τ−(2 k +1) .
Hyperbolic Lucas cosine
(5.18)
(5.19)
cL( x ) = τ2 k + τ−2 k .
Note that for the discrete values x=k the hyperbolic Fibonacci and Lucas
functions (5.16)(5.19) coincide with the Fibonacci and Lucas numbers, that is,
sF ( k ) = F2 k ; cF ( k ) = F2 k +1 ; sL ( k ) = L2k +1 ; cL ( k ) = L2k .
(5.20)
This means that the extended Fibonacci and Lucas numbers coincide with
the hyperbolic Fibonacci and Lucas functions for the discrete points of the
continuous variable x=k (k=0,±1,±2,±3,…), that is, the extended Fibonacci
and Lucas numbers are “discrete analogs” of the hyperbolic Fibonacci and
Lucas sine and cosine. This property (5.20) is a rather characteristic peculiarity
of the above hyperbolic Fibonacci and Lucas functions (5.16)(5.19) in compar
ison with the classical hyperbolic functions (5.7), (5.8) that, however, do not
have “discrete analogs” similar to (5.20).
Alexey Stakhov
THE MATHEMATICS OF HARMONY
266
5.3.2. The Hyperbolic Fibonacci and Lucas Tangent and Cotangent
Let us introduce the definitions of the hyperbolic Fibonacci and Lucas tangents
and cotangents by analogy to the classical hyperbolic tangent and cotangent.
Hyperbolic Fibonacci tangent:
sFx
tFx =
.
(5.21)
cFx
Taking into consideration (5.16) and (5.17), the hyperbolic Fibonacci tan
gent can be represented as follows:
tFx =
τ2 x − τ −2 x
(τ4 x − 1)τ2 x +1 τ(τ4 x − 1)
= 2 x 4 x +2
=
.
− (2 x +1)
+τ
+ 1) τ4 x +1 + 1
τ
τ (τ
2 x +1
(5.21)
Hyperbolic Fibonacci cotangent:
cFx
.
sFx
Taking into consideration (5.16) and (5.17), the expression for the hyper
bolic Fibonacci cotangent can be represented as follows:
τ4 x + 2 + 1
ctFx =
.
(5.22)
τ(τ4 x − 1)
ctFx =
Hyperbolic Lucas tangent:
sLx τ4 x + 2 − 1
.
tLx =
=
cLx τ(τ4 x + 1)
(5.23)
Hyperbolic Lucas cotangent:
cLx τ(τ4 x + 1)
=
.
(5.24)
sLx τ4 x + 2 − 1
Thus, the main result that follows from our study is the introduction of two
new kinds of elementary functions, the Hyperbolic Fibonacci Functions and the
Hyperbolic Lucas Functions. By their form, these functions are similar to the
classical hyperbolic functions, but differ from them by one important feature. In
contrast to the classical hyperbolic functions, the hyperbolic Fibonacci and Lu
cas functions have numerical analogs the classical Fibonacci and Lucas num
bers. In particular, the Fibonacci numbers with the even indices are discrete
analogs for the hyperbolic Fibonacci sine (5.16), the Fibonacci numbers with
the odd indices are discrete analogs for the hyperbolic Fibonacci cosine (5.17),
the Lucas numbers with the odd indices are discrete analogs for the hyper
bolic Lucas sine (5.18), and the Lucas numbers with the even indices are
discrete analogs for the hyperbolic Fibonacci cosine (5.19).
ctLx =
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
267
5.3.3. Some Properties of the Hyperbolic Fibonacci and Lucas Functions
5.3.3.1. Hyperbolic Fibonacci Sine
The function y=sF(x) is odd function because
(
sF ( − x ) = τ−2 x − τ−( −2 x )
)
(
5 = − τ2 x − τ−2 x
)
5 = − sF ( x ) .
Its value at the point x=0 is equal to
(
sF ( 0 ) = τ0 − τ0
)
5 = 0.
5.3.3.2. Hyperbolic Fibonacci Cosine
The function y=sF(x) is given by (5.17). Let us find the intersec
tion points of the function graph with the coordinate axes. As
cF ( x ) = τ2 x +1 + τ −(2 x +1)
5 > 0 for any x∈{∞,+∞} this means that the graph of
the function does not intersect the xaxes at any point.
For x=0 we have
(
(
cF ( 0 ) = τ1 + τ−1
)
)
5 = 1,
that is, the graph of the function y=sF(x) intersects the yaxis at the point y=1.
Now, let us examine the function y=cF(x) on the extremum; for this pur
pose we find its first derivative:
2 ln τ 2 x +1 −(2 x +1)
2
y′ = (cFx )′ =
τ
−τ
=
ln τ sLx .
5
5
Note that y′=0 for the condition:
(
)
τ2 x +1 − τ ( ) = 0; τ4 x + 2 = 1; 4 x + 2 = 0; x = −1 2 .
Thus, we have the extremum at the point with abscissa x=1/2.
It is easy to prove that the function y=cF(x) is symmetric about the line
x=1/2 because
− 2 x +1
cF  x − (1 2 ) = ( τ2 x + τ −2 x )
5 = cF  − x − (1 2 ) .
It is important to note that the function y=cF(x) is not symmetric about
the yaxis.
5.3.3.3. Hyperbolic Lucas Sine
The hyperbolic Lucas sine y=sL(x) is given by (5.18). Let us calculate the
value of the function at the point x=0:
(
sL ( 0 ) = τ − τ−1 = 1 + 5
) 2 + (1 − 5 ) 2 = 1.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
268
Now, let us calculate the value of x where the function y=sL(x) is equal to 0:
τ2 x +1 − τ ( ) = 0; τ4 x + 2 = 1; 4 x + 2 = 0; x = −1 2 .
As sL(1/2)=0, this means that the graph of the function y=sL(x) intersects
the xaxis at the point x = −1 2.
− 2 x +1
5.3.3.4. Hyperbolic Lucas Cosine
The function y=cL(x) is given by (5.19). By using the calculus methods
[154], we can find the following properties of the function y = cL(x). The
function is an even function because
cL ( − x ) = τ−2 x + τ ( ) = τ−2 x + τ2 x = cLx .
Let us find the intersection points of the function graph with the coordinate
axes. As cL(x)=τ2x+τ2x>0 for any x∈{∞,+∞}, this means that the graph of the
function does not intersect the xaxes at any point.
For the case x=0 we have: cL(0)=τ0+τ0=2; therefore, the graph of the func
tion intersects the yaxis at the point y=2.
Let us examine the function y=cL(x) on the extremum; with this purpose
we find the first derivative of the function:
y′ = cL ( x ) ′ = 2 ln τ ( τ2 x − τ −2 x ) .
− −2 x
The first derivative y′ turns out to be 0 only under the condition that
τ2xτ2x=0, this is possible only for the case x=0. This means that at the point x=0
the function has an extremum. The extreme value of the function y=cL(x) at the
point x=0 is equal to cL(0)=2.
By analogy, we can examine other hyperbolic Fibonacci and Lucas functions,
in particular, tangent and cotangent, secant and cosecant.
5.4. Integration and Differentiation of the Hyperbolic Fibonacci and Lucas
Functions and their Main Identities
5.4.1. Integration of the Function y=sF(x)
Represent the function y=sF(x) given by (5.16) in the form:
sF ( x ) = ( τ2 x − τ −2 x ) 5 = ( e2 x ln τ − e −2 x ln τ ) 5 .
(5.25)
Then, by using the representation (5.25), we obtain the following expres
sion for the integral of the function y=sF(x):
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
269
∫ sF ( x ) dx = ∫ ( e − e
= (1 5 ) ( ∫ e
dx − ∫ e
2 x ln τ
2 x ln τ
−2 x ln τ
−2 x ln τ
)
dx
)
5  dx

( 5 ) (1 2 ln τ ) e + (1 2 ln τ) e 
= (1 2 5 ln τ ) ( e
+e
)
= (1 2 5 ln τ ) ( τ + τ )
= (1 2 5 ln τ ) cLx = (1 2 ln τ ) cF  x − (1 2 ) .
2 x ln τ
= 1
2 x ln τ
−2 x ln τ
−2 x ln τ
−2 x
2x
5.4.2. Integration of the Function y=cF(x)
By using the above approach, we obtain the following expression for the in
tegral of the function y=cF(x) given by (5.17):
∫ cF ( x ) dx = ∫ ( e( ) + e ( ) ) 5  dx
= (1 5 ) ( ∫ e ( ) dx + ∫ e ( ) dx )
2 x +1 ln τ
− 2 x +1 ln τ
2 x +1 ln τ
− 2 x +1 ln τ
( 5 ) (1 2 ln τ ) e( ) − (1 2 ln τ ) e ( ) 
= (1 2 5 ln τ ) ( e ( ) − e ( ) )
= (1 2 5 ln τ ) ( τ
−τ ( ))
= (1 2 5 ln τ ) sLx = (1 2 ln τ ) sF  x + (1 2 ) .
2 x +1 ln τ
= 1
2 x +1 ln τ
2 x +1
− 2 x +1 ln τ
− 2 x +1 ln τ
− 2 x +1
5.4.3. Integration of the Function y=sL(x)
∫ sL ( x ) dx = ∫ ( e
( 2 x +1)ln τ
− e −(2 x +1)ln τ ) dx
= (1 2 ln τ ) e(2 x +1)ln τ + (1 2 ln τ ) e −(2 x +1)ln τ
= (1 2 ln τ )  e(2 x +1)ln τ + e −(2 x +1)ln τ 
= (1 2 ln τ )  τ2 x +1 + τ −(2 x +1) 
=
( 5 2 ln τ ) cFx = (1 2 ln τ ) cL  x + (1 2) .
5.4.4. Integration of the Function y=cL(x)
∫ cL ( x ) dx = ∫ ( e
2 x ln τ
+ e −2 x ln τ ) dx
= (1 2 ln τ ) e2 x ln τ − (1 2 ln τ ) e −2 x ln τ
= (1 2 ln τ )  e2 x ln τ − e −2 x ln τ 
= (1 2 ln τ )  τ2 x − τ −2 x 
=
( 5 2 ln τ ) sFx = (1 2 ln τ ) sL  x − (1 2) .
Alexey Stakhov
THE MATHEMATICS OF HARMONY
270
The results of integration are presented in Table 5.1.
Table 5.1. Formulas for the integrals of the hyperbolic Fibonacci and Lucas functions
sF ( x )
y
1
∫ ydx
cL ( x ) =
5 ln τ
2
1
cF
2 ln τ
( )
x −
cF ( x )
1
2
1
5 ln τ
2
sL ( x ) =
sL ( x )
y
5
∫ ydx
2 ln τ
cF ( x ) =
1
sF
2 ln τ
( )
x +
1
2
cL ( x )
1
2 ln τ
( )
cL x +
1
5
2
2 ln τ
sF ( x ) =
1
2 ln τ
( )
sL x −
1
2
5.4.5. Differentiation of the Function y=sF(x)
By using the expression (5.25), we obtain the following expressions for the
derivatives of the function y=sF(x):
y′ = ( 2 ln τ e2 x ln τ + 2 ln τ e −2 x ln τ )
(
= 2 ln τ
)
(
5 ( τ2 x + τ −2 x ) = 2 ln τ
(
5
)
5 cL ( x ) ;
)
y″ = 2 ln τ
5 ( 2 ln τ e2 x ln τ − 2 ln τ e −2 x ln τ )
= ( 2 ln τ )

5  ( τ2 x − τ −2 x ) = ( 2 ln τ ) sF ( x ) ;

2
2
(
2
y′′′ = ( 2 ln τ )

= ( 2 ln τ )

3
(
5  2 ln τ e2 x ln τ + 2 ln τ e −2 x ln τ

5  τ 2 x + τ −2 x

)
)
3
2
= ( 2 ln τ )
5  cL ( x ) = ( 2 ln τ ) y′;


...
2 k +1
2 ln τ )
(
2k
(2 k +1)
y
=
cL ( x ) = ( 2 ln τ ) y′;
5
2k
(2k )
y
= ( 2 ln τ ) sF ( x ) .
5.4.6. Differentiation of the Function y=cF(x)
By representing the function y=cF(x) in the form
y = cF ( x ) =  τ(

= e(

2 x +1) ln τ
2 x +1)
+e (
+τ (
− 2 x +1) 
− 2 x +1) ln τ 


5
5
(5.26)
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
271
and repeatedly carrying out the differentiation of (5.26), we obtain the following
expressions for the derivatives of the function y=cF(x):
(
)
5 sL ( x ) ;
y′ = 2 ln τ
y'' = ( 2 ln τ ) cF ( x ) ;
2
3
2
y''' = ( 2 ln τ )
5  sL ( x ) = ( 2 ln τ ) y′;


4
IV
y ( ) = ( 2 ln τ ) cF ( x ) ;
...
2 k +1
2k
y(2 k +1) = ( 2 ln τ )
5  sL( x ) = ( 2 ln τ ) y′;


2k
2k
y ( ) = ( 2 ln τ ) cF ( x ) .
5.4.7. Differentiation of the Function y=sLx
By representing the function y=sLx given by (5.18) in the form
y = sL ( x ) = τ2 x +1 − τ−(2 x +1) = e(
2 x +1) ln τ
−e (
− 2 x +1) ln τ
,
(5.27)
and repeatedly carrying out the differentiation of (5.27), we obtain the following
expressions for the derivatives of the function y=sLx:
y′ = 2 5 ln τ cF ( x ) ;
y″ = (2 ln τ )2 sL ( x ) ;
y′′′ = (2 ln τ )3 5cF ( x ) = ( 2 ln τ ) y′;
2
IV
y ( ) = (2 ln τ )4 sL ( x ) ;
y ( ) = (2 ln τ )5 5cF ( x ) = ( 2 ln τ ) y′;
...
4
V
y(
2 k +1)
= 5 ( 2 ln τ )
2 k +1
cF ( x ) = ( 2 ln τ ) y′;
2k
2k
2k
y ( ) = ( 2 ln τ ) sL ( x ) .
5.4.8. Differentiation of the Function y=cLx
By representing the function y=cLx in the form
(5.28)
y = cL ( x ) = τ2 x + τ−2 x = e 2 x ln τ + e −2 x ln τ
and repeatedly carrying out the differentiation of (5.28), we obtain the following
expressions for the derivatives of the function y=cLx:
y′ = 2 5 ln τ sF ( x ) ;
y″ = ( 2 ln τ ) cL ( x ) ;
2
Alexey Stakhov
THE MATHEMATICS OF HARMONY
272
y = 5 ( 2 ln τ ) sF ( x ) = ( 2 ln τ ) y′;
3
2
y ( IV ) = ( 2 ln τ ) cL ( x ) ;
4
y(V ) = 5 ( 2 ln τ ) sF ( x ) = ( 2 ln τ ) y′;
...
5
4
y ( 2k +1) = 5 ( 2 ln τ )
2 k +1
sF ( x ) = ( 2 ln τ ) y′;
2k
y (2 k ) = ( 2 ln τ ) cL ( x ) .
2k
The results of differentiation are given in Table 5.2.
Table 5.2. Formulas for the derivatives of hyperbolic Fibonacci and Lucas functions
y
y′
sF ( x )
2 ln τ
cL ( x )
5
y
y
2 ln τ
sL ( x )
sL ( x )
2
5 ln τcF ( x )
cL ( x )
2
5 ln τsF ( x )
5
y ′′
( 2 ln τ )2 sF ( x )
( 2 ln τ )2 cF ( x )
( 2 ln τ )2 sL ( x )
( 2 ln τ )2 cL ( x )
y ′′′
( 2 ln τ )2 y ′
( 2 ln τ )2 y ′
( 2 ln τ )2 y ′
( 2 ln τ )2 y ′
( IV )
( 2 ln τ )4 sF ( x )
( 2 ln τ ) 4 cF ( x )
( 2 ln τ ) 4 sL ( x )
( 2 ln τ ) 4 cL ( x )
(V )
( 2 ln τ ) 4 y ′
( 2 ln τ ) 4 y ′
( 2 ln τ )4 y ′
( 2 ln τ )4 y ′
#
#
#
#
#
y
y
cF ( x )
(2k )
( 2 k +1 )
( 2 ln τ )2 k sF ( x ) ( 2 ln τ )2 k cF ( x ) ( 2 ln τ )2 k sL ( x ) ( 2 ln τ )2 k cL ( x )
( 2 ln τ )2 k y ′
( 2 ln τ )2 k y ′
( 2 ln τ )2 k y ′
( 2 ln τ )2 k y ′
5.4.9. The Main Identities for the Hyperbolic Fibonacci and Lucas
Functions
The hyperbolic Fibonacci and Lucas functions (5.16)(5.19) are a generaliza
tion of the classical Fibonacci and Lucas numbers given by Binet formulas (5.12)
(5.15). These hyperbolic Fibonacci and Lucas functions (5.16)(5.19) are connected
with the classical Fibonacci and Lucas numbers through simple correlations (5.20).
Using a geometric representation of the hyperbolic Fibonacci and Lucas functions,
we result in a very simple geometric interpretation of the correlations (5.20). The
classical Fibonacci and Lucas numbers are as if inscribed into the graphs of the
hyperbolic Fibonacci and Lucas functions at the discrete points of the variable
x=0,±1,±2,±3,… .Thus, the hyperbolic Fibonacci and Lucas functions are an ex
tension of the classical Fibonacci and Lucas numbers for the continuous domain.
From this it appears that the hyperbolic Fibonacci and Lucas functions possess
recursive properties similar to those of the classical Fibonacci and Lucas numbers.
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
273
On the other hand, functions (5.16)(5.19) are similar to the classical hyper
bolic functions (5.7) and (5.8). We can assume by analogy that the hyperbolic
Fibonacci and Lucas functions possess hyperbolic properties similar to the prop
erties of classical hyperbolic functions.
Let us examine these analogies in greater detail, starting with the sim
plest recursive relation that gives classical Fibonacci numbers:
Fn+1= Fn + Fn1.
(5.29)
If we accept n=2k or n=2k+1, where k=0,±1,±2,±3,…, we can rewrite the
recursive relation (5.29) in two ways:
F2k+1= F2k + F2k1
( 5.30)
F2k+2=F2k+1+F 2k.
(5.31)
Using the correlations (5.20), we can rewrite the recursive relations (5.30)
and (5.31) in terms of hyperbolic Fibonacci and Lucas functions as follows:
cF(k)=sF(k)+cF(k1),
(5.32)
sF(k+1)=cF(k)+sF(k),
(5.33)
where k is a discrete variable that takes its values from the set {0,±1,±2,±3,…}.
Substituting the discrete variable k with the continuous variable x in the
recursive relations (5.32) and (5.33), we obtain two important identities for
the hyperbolic Fibonacci and Lucas functions:
cF(x)=sF(x)+cF(x1)
(5.34)
sF(x+1)=cF(x)+sF(x).
(5.35)
Let us represent the recursive relation Ln+1= Ln + Ln1 for the classical Lucas
numbers in the form of two recursive relations
L2k+1=L2k +L2k1
(5.36)
L2k+2=L2k+1+L2k,
(5.37)
where k is a discrete variable that takes its values from the set {0,±1,±2,±3,…}.
Then we represent (5.36) and (5.37) in terms of the hyperbolic Fibonacci and
Lucas functions, rewriting the recursive relations (5.36) and (5.37) as follows:
sL(k)=cL(k)+sL(k1)
(5.38)
cL(k+1)=sL(k)+cL(k).
(5.39)
Substituting the continuous variable x for the discrete variable k in the re
cursive relations (5.38) and (5.39), we obtain two important identities:
sL(x)=cL(x)+sL(x1)
(5.40)
cL(x+1)=sL(x)+cL(x).
(5.41)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
274
Note that there is another way to obtain the identities (5.34), (5.35), (5.40),
and (5.41). For example, let us prove the identity (5.35) by using the definitions
(5.16) and (5.17):
sF ( x ) + cF ( x ) = ( τ2 x − τ −2 x )
=  τ2 x ( τ + 1) + τ −2 x ( τ −1 − 1)
5 +  τ2 x +1 + τ −(2 x +1) 
5
5 =  τ2( x +1) − τ −2( x +1) 
5 = sF ( x + 1) .
By analogy we can prove the other identities (5.34), (5.40) and (5.41).
The above examination has very unusual consequences for the Fibonacci
number theory [13, 16, 28]. We can assume the following hypothesis: all well
known identities for the Fibonacci and Lucas numbers have the “hyperbolic in
terpretations” in the form of corresponding identities for the hyperbolic Fibonacci
and Lucas functions. For example, let us give the “hyperbolic interpretation” of
the famous Cassini formula that connects three adjacent Fibonacci numbers:
(5.42)
Fn2 − Fn −1Fn +1 = ( −1) ,
where n=0,±1,±2,±3,… is a discrete variable.
We can represent the Cassini formula (5.42) in the form of two formulas for
the even (n=2k) and odd (n=2k+1) values of the discrete variable n as follows:
n +1
F22k − F2 k −1F2 k +1 = −1
(5.43)
(5.44)
F22k +1 − F2 k F2 k + 2 = 1,
where k is a discrete variable that takes its values from the set {0,±1,±2,±3,…}.
By using (5.20), we can represent the identities (5.43) and (5.44) as follows:
sF 2 ( k ) − cF ( k − 1) cF ( k ) = −1
(5.45)
sF 2 ( x ) − cF ( x − 1) cF ( x ) = −1
(5.47)
(5.46)
cF 2 ( k ) − sF ( k ) sF ( k + 1) = 1.
Substituting the continuous variable x for the discrete variable k in the for
mulas (5.45) and (5.46), we obtain two important identities for the hyperbolic
Fibonacci and Lucas functions:
(5.48)
cF 2 ( x ) − sF ( x ) sF ( x + 1) = 1.
We can prove the identities (5.47) and (5.48) by using the definitions (5.16)
and (5.17). For example, let us prove the identity (5.48):
(( τ
(
2 x +1
+ τ −(2 x +1) ) / 5
) − (( τ − τ ) / 5 ) × (( τ
2
2x
−2 x
) (
2 x +2
− τ −(2 x +2) ) / 5
)
= ( τ4 x +2 + 2τ2 x +1τ −(2 x +1) + τ −(4 x +2) ) / 5 − ( τ4 x +2 − τ2 − τ −2 + τ −(4 x +2) ) / 5
= ( 2 + τ2 + τ −2 ) / 5.
)
(5.49)
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
275
In Chapter 2 we derived the Binet formulas for Lucas numbers given by
(2.67). For the even n=2k the formula (2.67) can be written as follows:
L2k=τ2k+τ2k.
For k=1 we have:
(5.50)
L 2=τ 2+τ 2=3.
(5.51)
Substituting (5.51) into (5.49), we obtain the identity (5.48).
It is important to note that the hyperbolic Fibonacci and Lucas functions
are a generalization and extension of the Fibonacci and Lucas numbers for the
continuous domain. This means that the classical Fibonacci number theory [13,
16, 28] is reducible to a more general theory the theory of the hyperbolic Fi
bonacci and Lucas functions. All identities for the classical Fibonacci and Lucas
numbers have their continuous analogs in the form of the corresponding identi
ties for the hyperbolic Fibonacci and Lucas functions, and conversely.
As practical examples for students we can consider the following
“recursive” identities for the hyperbolic Fibonacci and Lucas functions:
1
sF ( x ) cF ( x ) =  sL ( 2 x ) − 1
5
1

sF ( x ) sL ( x ) = 5sF ( x ) sF  x + 
2

1

cF ( x ) cL ( x ) = 5 cF ( x ) cF  x − 
2

1
1

sF ( x ) =
sL  x − 
2
5 
1 1

cF  x −  =
cL ( x )
2
5


 2y −1  
 2x + 1 
+ sL2 
cF ( x ) + cF ( y ) = 5  cL2 


 4 
 4 

x
sF ( 3 x ) = sF   cL ( x ) + 1
2
x
cL ( 3 x ) = sF   cL ( x ) + 1
2
Let us next consider some “hyperbolic” properties of the hyperbolic Fi
bonacci and Lucas functions. Consider these important properties for the
classical hyperbolic functions:
Alexey Stakhov
THE MATHEMATICS OF HARMONY
276
Evenness property
sh ( − x ) = − sh ( x ) ; ch ( − x ) = ch ( x ) ; th ( − x ) = th ( x ) .
Formulas for addition
sh ( x + y ) = sh ( x ) ch ( y ) + sh ( y ) ch ( x ) ;
ch ( x + y ) = ch ( x ) ch ( y ) + sh ( y ) sh ( x ) .
Formulas for the double angle
sh ( 2 x ) = 2ch ( x ) sh ( x ) ; ch ( 2 x ) = 2ch 2 ( x ) − 1
It is easy to prove that the hyperbolic Fibonacci and Lucas functions possess
similar “hyperbolic” properties, for example,
sF ( 2 x ) = sF ( x ) cL ( x )
cF ( 2 x + 1) = cF ( x ) sL ( x )
sF ( x ) sL ( x ) = sF ( 2 x ) − 1
cF ( x ) cL ( x ) = cF ( 2 x ) + 1
sL ( 2 x ) = sL ( x ) cL ( x ) + 1
1
sF ( y ) sF ( x ) =  sL ( x + y ) − sL ( x − y ) 
5
sL ( x ) cL ( x ) = sL ( x + y ) + sL ( x − y )
 x+y  x−y
sF ( x ) + sF ( y ) = sF 
 cL 

 2   2 
 x−y  x+y
sF ( x ) − sF ( y ) = sF 
 cL 

 2   2 
x+y  x−y
cF ( x ) + cF ( y ) = cF 
 cL 

 2   2 
 x−y  x+y
cF ( x ) − cF ( y ) = sF 
 sL 

 2   2 
ctF ( 2 x ) =
sF 2 ( x ) + cL2 ( x )
2 sF ( x ) cL ( x )
The above formulas produce only a small portion of the huge number of math
ematical identities for the hyperbolic Fibonacci and Lucas functions. Their proof
can become a fascinating pastime for students interested in Fibonacci mathe
matics and its applications.
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
277
5.5. Symmetric Hyperbolic Fibonacci and Lucas Functions (Stakhov Rozin
Definition)
The above hyperbolic Fibonacci and Lucas functions given by (5.16)(5.19)
have an essential shortcoming relative to the classical hyperbolic functions. In par
ticular, they don’t possess the evenness property of classical hyperbolic functions:
sh ( − x ) = − sh ( x ) ; ch ( − x ) = ch ( x ) ; th ( − x ) = th ( x ) .
(5.52)
To overcome this shortcoming, Stakhov and Rozin introduced [106] the so
called Symmetric Hyperbolic Fibonacci and Lucas Functions:
Symmetric hyperbolic Fibonacci sine
sFs ( x ) =
τ x − τ− x
cFs ( x ) =
τ x + τ− x
5
Symmetric hyperbolic Fibonacci cosine
5
Symmetric hyperbolic Lucas sine
sLs ( x ) = τ x − τ− x
Symmetric hyperbolic Lucas cosine
(5.53)
(5.54)
(5.55)
(5.56)
cLs ( x ) = τ x + τ− x
The Fibonacci and Lucas numbers are determined identically by the sym
metric hyperbolic Fibonacci and Lucas functions as follows:
 sFs ( n ) , for n = 2k
Fn = 
cFs ( n ) , for n = 2k + 1
 cLs ( n ) , for n = 2k
.
Ln = 
 sLs ( n ) , for n = 2k + 1
(5.57)
It is easy to prove that the function (5.53) is an odd function because
sFs ( − x ) =
τ− x − τ x
5
=−
τ x − τ− x
5
= − sFs ( x ) .
(5.58)
On the other hand,
cFs ( − x ) =
τ− x + τ x
5
= cFs ( x ) ,
(5.59)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
278
that is, the symmetric hyperbolic Fibonacci cosine (5.54) is an even func
tion. By analogy we can prove the following properties of the hyperbolic Lucas
functions:
sL ( − x ) = − sL ( x )
(5.60)
cL ( − x ) = cL ( x ) .
(5.61)
The properties (5.58)(5.61) show that the symmetric hyperbolic Fibonacci
and Lucas functions (5.52)(5.55) possess the evenness property (5.52).
It is easy to construct the graphs of symmetric hyperbolic Fibonacci and
Lucas sines and cosines (Fig. 5.3 and Fig. 5.4). Their graphs have symmetric
form and in this respect are similar to the classical hyperbolic functions.
Y
Y
y = cLs(x)
y = cFs(x)
1
1
y = sFs(x)
0
1
Figure 5.3. A graph of the symmetric
hyperbolic Fibonacci sine and cosine
y = sLs(x)
0
1
Figure 5.4. A graph of the symmetric
hyperbolic Lucas sine and cosine
Here it is necessary to point out that at x=0 the symmetric hyperbolic Fibonac
ci cosine cFs(x) takes the value cFs(0) = 2 5 , and the symmetric hyperbolic Lucas
cosine cLs(x) takes the value cLs(0)=2. It is also important to emphasize that the
Fibonacci numbers Fn with even indices (n=0,±2,±4,±6,…) are “inscribed” into the
graph of symmetric hyperbolic Fibonacci sine sFs(x) at the discrete points
(x=0,±2,±4,±6,…) and the Fibonacci numbers with odd indices (n=±1,±3,±5,…)
are “inscribed” into the symmetric hyperbolic Fibonacci cosine cFs(x) at the dis
crete points (x=±1,±3,±5,…). On the other hand, the Lucas numbers Ln with the
even indices are “inscribed” into the graph of symmetric hyperbolic Lucas cosine
cLs(x) at the discrete points (n=0,±2,±4,±6,…) and the Lucas numbers with odd
indices are “inscribed” into the graph of the symmetric hyperbolic Lucas cosine sLs(x)
at the discrete points (x=±1,±3,±5,…).
The symmetric hyperbolic Fibonacci and Lucas functions are connected
amongst themselves by the following simple correlations:
Chapter 5
sFs ( x ) =
Hyperbolic Fibonacci and Lucas Functions
279
sLs ( x )
; cFs ( x ) =
cLs ( x )
.
5
5
Also we can introduce the notions of symmetric hyperbolic Fibonacci and
Lucas tangents and cotangents.
Symmetric hyperbolic Fibonacci tangent
tFs ( x ) =
sFs ( x )
cFs ( x )
=
τ x − τ− x
τ x + τ− x
(5.62)
Symmetric hyperbolic Fibonacci cotangent
ctFs ( x ) =
cFs ( x )
sFs ( x )
=
τ x + τ− x
τ x − τ− x
(5.63)
Symmetric hyperbolic Lucas tangent
tLs ( x ) =
sLs ( x )
cLs ( x )
=
τ x − τ− x
τ x + τ− x
(5.64)
Symmetric hyperbolic Lucas cotangent
ctLs ( x ) =
cLs ( x )
sLs ( x )
=
τ x + τ− x
τ x − τ− x
(5.65)
We conclude from a comparison of the functions (5.62) with (5.63), and (5.64)
with (5.65) that these functions coincide, that is, we have:
tFs ( x ) = tLs ( x ) and ctFs ( x ) = ctLs ( x ) .
(5.66)
It is easy to prove that the functions (5.62) and (5.63) are odd functions
because
τ− x − τ x
τ x − τ− x
tFs ( − x ) = − x
=
−
= −tFs ( x )
τ + τx
τ x + τ− x
τ− x + τ x
τ x + τ− x
ctFs ( − x ) = − x
=
−
= −ctFs ( x ) .
τ − τx
τ x − τ− x
Taking into consideration (5.66), we can write:
tLs ( − x ) = − tLs ( x ) ; ctLs ( − x ) = −ctLs ( x ) ,
that is, the functions (5.64) and (5.65) are also odd functions.
Next let us compare the classical hyperbolic functions (5.7), (5.8) with
the symmetric hyperbolic Fibonacci functions (5.53)(5.56). It follows from
this comparison that the symmetric hyperbolic Fibonacci and Lucas func
tions possess all the important properties of classical hyperbolic functions.
Thus, the symmetric hyperbolic Fibonacci and Lucas functions have Hy
perbolic Properties.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
280
On the other hand, a comparison of Fibonacci and Lucas numbers with the
symmetric hyperbolic Fibonacci and Lucas functions show that according to
(5.61) these functions are a generalization of Fibonacci and Lucas numbers for
the continuous domain. This means that the symmetric hyperbolic Fibonacci
and Lucas functions possess Recursive Properties similar to the properties of Fi
bonacci and Lucas numbers.
5.6. Recursive Properties of the Symmetric Hyperbolic Fibonacci and
Lucas Functions
Consider the recursive properties of the symmetric hyperbolic Fibonacci
and Lucas functions in comparison with the analogous properties of Fibonac
ci and Lucas numbers.
Theorem 5.1. The following correlations, that are analogous to the recursive
correlation for the Fibonacci numbers Fn+2=Fn+1+Fn, are valid for the symmetric
hyperbolic Fibonacci functions:
sFs ( x + 2 ) = cFs ( x + 1) + sFs ( x )
(5.67)
cFs ( x + 2 ) = sFs ( x + 1) + cFs ( x ) .
Proof:
cFs ( x + 1) + sFs ( x ) =
=
=
− x +1)
τ x ( τ + 1) − τ − x (1 − τ )
5
τ
x +2
−( x +2)
−τ
5
5
−( x +2)
τ x − τ− x
5
τ x × τ 2 − τ − x × τ −2
5
τ x +1 − τ (
5
τ x ( τ + 1) + τ − x (1 − τ )
x +2
=
+
= sFs ( x + 2) ;
sFs ( x + 1) + cFs ( x ) =
=
τ x +1 + τ (
5
− x +1)
=
+
τ x + τ− x
5
τ x × τ 2 + τ − x × τ −2
5
+τ
= cFs ( x + 2) .
5
Theorem 5.2. The following correlations, that are analogous to the recursive
equation for the Lucas numbers Ln+2=Ln+1+Ln, are valid for the symmetric hy
perbolic Lucas functions:
=
τ
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
281
sLs ( x + 2 ) = cLs ( x + 1) + sLs ( x )
(5.68)
cLs ( x + 2 ) = sLs ( x + 1) + cLs ( x ) .
The proof is analogous to Theorem 5.28.
Theorem 5.3 (a generalization of Cassini’s formula). The following corre
n +1
lations, that are similar to Cassini’s formula Fn2 − Fn +1Fn −1 = ( −1) , are valid
for the symmetric hyperbolic Fibonacci functions:
 sFs ( x )  − cFs ( x + 1) cFs ( x − 1) = −1
2
(5.69)
cFs ( x )  − sFs ( x + 1) sFs ( x − 1) = 1.
Proof:
2
 sFs ( x ) − cFs ( x + 1) cFs ( x − 1)
2
 τ x − τ − x  τ x +1 + τ −( x +1) τ x −1 + τ−( x −1)
=
×
 −
5
5 
5

2
=
τ2 x − 2 + τ −2 x − ( τ2 x + τ2 + τ −2 + τ −2 x )
5
= −1;
cFs ( x )  − sFs ( x + 1) sFs ( x − 1)
2
 τ x + τ − x  τ x +1 − τ −( x +1) τ x −1 − τ −( x −1)
=
×
 −
5 
5
5

2
=
(
τ2 x + 2 + τ−2 x − τ2 x − τ2 − τ−2 + τ−2 x
5
) = 1.
Note that for the proof we used the Binet formula for the Lucas numbers:
τ2 + τ−2 = L ( 3 ) = 3.
Theorem 5.4. The following correlations, that are similar to the identity
n
L2n − 2 ( −1) = L2 n , are valid for the symmetric hyperbolic Lucas functions:
 sLs ( x ) + 2 = cLs ( 2 x ) ; cLs ( x )  − 2 = sLs ( 2 x ) .
(5.70)
The proof is analogous to Theorem 5.3.
Theorem 5.5. The following correlations, that are similar to the iden
tity F n+1+F n1=Ln, are valid for the symmetric hyperbolic Fibonacci and
Lucas functions:
2
cFs ( x + 1) + cFs ( x − 1) = cLs ( x )
sFs ( x + 1) + sFs ( x − 1) = sLs ( x ) .
2
(5.71)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
282
The proof is analogous to Theorem 5.1.
Theorem 5.6. The following correlations, that are similar to the identity
Fn+L n=2Fn+1, are valid for the symmetric hyperbolic Fibonacci and Lucas
functions:
cFs ( x ) + sLs ( x ) = 2sFs ( x + 1)
(5.72)
sFs ( x ) + cLs ( x ) = 2cFs ( x ) .
The proof is analogous to Theorem 5.1.
Based on the definitions (5.53) (5.56), we can prove different identities
for the symmetric hyperbolic Fibonacci and Lucas functions. We can see
some of these identities in Table 5.3.
Table 5.3. The identities for Fibonacci and Lucas numbers
and for symmetric hyperbolic Fibonacci and Lucas functions
Fn + 2 = Fn +1 + Fn
Ln + 2 = Ln +1 + Ln
Fn = ( −1)
n +1
F− n
sFs ( x + 2 ) = cFs ( x + 1) + sFs ( x )
cFs ( x + 2 ) = sFs ( x + 1) + cFs ( x )
sLs ( x + 2 ) = cLs ( x + 1) + sLs ( x )
cLs ( x + 2 ) = sLs ( x + 1) + cLs ( x )
sFs ( x ) = − sFs ( − x )
cFs ( x ) = cFs ( − x )
sLs ( x ) = sLs ( − x )
cLs ( x ) = − cLs ( − x )
sFs ( x + 3 ) + cFs ( x ) = 2cFs ( x + 2 )
cFs ( x + 3 ) + sFs ( x ) = 2sFs ( x + 2 )
sFs ( x + 6 ) − cFs ( x ) = 4 sFs ( x + 3 )
cFs ( x + 6 ) − sFs ( x ) = 4cFs ( x + 3 )
[ sFs ( x )] − cFs ( x + 1) cFs ( x − 1) = −1
[cFs ( x )]2 − sFs ( x + 1) sFs ( x − 1) = 1
F2 n +1 = Fn2+1 + Fn2
cFs ( 2 x + 1) = [cFs ( x + 1)] + [ sFs ( x )]
2
sFs ( 2 x + 1) = [ sFs ( x + 1)] + [cFs ( x )]
L2n − 2 ( −1) = L2 n
Ln + 5 Fn = 2 Ln +1
[ sLs ( x )]2 + 2 = cLs ( 2 x )
sLs ( x ) + cLs ( x + 3 ) = 2 sLs ( x + 2 )
2
sLs ( x + 1) sLs ( x − 1) − [ cLs ( x )] = −5
sFs ( x + 3 ) − 2cFs ( x ) = sLs ( x )
sLs ( x − 1) + sLs ( x + 1) = 5 sFs ( x )
sLs ( x ) + 5cFs ( x ) = 2cLs ( x + 1)
[cLs ( x )]2 − 2 = cLs ( 2 x )
cLs ( x ) + sLs ( x + 3 ) = 2cLs ( x + 2 )
2
cLs ( x + 1) cLs ( x − 1) − [ sLs ( x )] = 5
cFs ( x + 3 ) − 2 sFs ( x ) = cLs ( x )
cLs ( x − 1) + cLs ( x + 1) = 5cFs ( x )
cLs ( x ) + 5sFs ( x ) = 2sLs ( x + 1)
L2n +1 + L2n = 5 F2 n +1
[ sLs ( x + 1)]2 + [cLs ( x )]2 = 5cFs ( 2 x + 1) [cLs ( x + 1)]2 + [ sLs ( x )]2 = 5sFs ( 2 x + 1)
Ln = ( −1) L− n
n
Fn + 3 + Fn = 2Fn +2
sFs ( x + 3 ) − cFs ( x ) = 2 sFs ( x + 1)
Fn + 3 − Fn = 2Fn +1
Fn +6 − Fn = 4 Fn + 3
Fn2 − Fn +1Fn −1 =
( −1 )
n +1
Ln + Ln + 3 = 2 Ln + 2
Ln +1Ln −1 − L2n = −5 ( −1)
Fn + 3 − 2 Fn = Ln
Ln −1 + Ln +1 = 5 Fn
2
2
n
n
cFs ( x + 3 ) − sFs ( x ) = 2cFs ( x + 1)
2
2
We can see from Table 5.3 that two identities for hyperbolic Fibonacci and
Lucas functions correspond to one identity for Fibonacci and Lucas numbers.
This fact can be explained very easily. This all depends on the evenness of the
index n of the Fibonacci and Lucas numbers. For example, consider the sim
plest identity Fn+2 = Fn+1+Fn. If n=2k (n is even), we should use the first identi
ty sFs(x+2)= cFs(x+1)+sFs(x); in the opposite case (n=2k+1) we should use
the other identity cFs(x+2)= sFs(x+1)+cFs(x).
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
283
5.7. Hyperbolic Properties of the Symmetric Hyperbolic Fibonacci and Lu
cas Functions and Formulas for Their Differentiation and Integration
5.7.1. Hyperbolic Properties
The symmetric hyperbolic Fibonacci and Lucas functions possess “hyper
bolic” properties similar to classical hyperbolic functions. Consider some of them
in contrast to certain properties of classical hyperbolic functions.
Theorem 5.7. The following identity, that is similar to the identity [ch(x)]2
[ch(x)]2 =1, is valid for the symmetric hyperbolic Fibonacci function:
2
2
4
cFs ( x ) −  sFs ( x )  = .
(5.73)
5
Proof:
2
 τ x + τ− x   τ x − τ− x 
cFs ( x )  −  sFs ( x )  = 
 −

5  
5 

2
2
2
τ2 x + 2 + τ−2 x − τ2 x + 2 − τ−2 x 4
= .
5
5
By analogy, we can prove the following theorem for symmetric hyperbolic
Lucas functions.
Theorem 5.8.
=
cLs ( x ) −  sLs ( x ) = 4.
(5.74)
Theorem 5.9. The following identity, that is similar to the identity
ch(x+y)=ch(x)ch(y)+sh(x)sh(y), is valid for the symmetric hyperbolic
Fibonacci function:
2
cFs ( x + y ) = cFs ( x ) cFs ( y ) + sFs ( x ) sFs ( y ) .
(5.75)
5
Proof:
2
2
τ x + τ− x τ y + τ− y τ x − τ− x τ y − τ− y
×
+
×
5
5
5
5
x+y
x−y
−x+y
− x−y
x+y
x−y
−x+y
− x−y
τ +τ +τ
+τ
+τ −τ −τ
+τ
=
5
cFs ( x ) cFs ( y ) + sFs ( x ) sFs ( y ) =
=
2 ( τ x + y + τ− x − y )
5
=
2
cFs ( x + y ) .
5
Alexey Stakhov
THE MATHEMATICS OF HARMONY
284
Theorem 5.10. The identity, that is similar to the identity ch(xy)=ch(x)ch(y)
sh(x)sh(y), is valid for the symmetric hyperbolic Fibonacci function:
2
cFs ( x − y ) = cFs ( x ) cFs ( y ) − sFs ( x ) sFs ( y ) .
(5.76)
5
By analogy we can prove the following theorems for the symmetric hyper
bolic Lucas functions.
Theorem 5.11.
2cLs ( x ± y ) = cLs ( x ) cLs ( y ) ± sLs ( x ) sLs ( y ) .
(5.77)
Theorem 5.12. The following identities, that are similar to the identity
ch(2x)=[ch(x)]2+[sh(x)]2, are valid for the symmetric hyperbolic Fibonacci and
Lucas functions:
2
2
2
cFs ( 2 x ) = cFs ( x )  +  sFs ( x )
(5.78)
5
2cLs ( 2 x ) = cLs ( x )  +  sLs ( x )  .
2
2
(5.79)
Theorem 5.13. The following identities, that are similar to the identity
sh(2x)=2sh(x)ch(x), are valid for the symmetric hyperbolic Fibonacci and
Lucas functions:
1
sFs ( 2 x ) = sFs ( x ) cFs ( x )
(5.80)
5
sLs ( 2 x ) = sLs ( x ) cLs ( x ) .
(5.81)
Theorem 5.14. The following formulas, that are similar to Moivre’s formu
las [ch(x)±sh(x)]n =ch(nx)±sh(nx), are valid for the symmetric hyperbolic Fi
bonacci and Lucas functions:
n −1
n
 2 
cFs ( x ) ± sFs ( x )  = 
(5.81)
  cFs ( nx ) ± sFs ( nx )
 5
cLs ( x ) ± sLs ( x ) = 2n −1 cFs ( nx ) ± sFs ( nx )  .
n
(5.82)
5.7.2. Formulas for Differentiation and Integration
It is easy to prove the following formulas for differentiation and integration
of symmetric hyperbolic Fibonacci and Lucas functions.
Theorem 5.15 (formulas for differentiation). The following correlations
similar to the nth derivatives of the classical hyperbolic functions
ch ( x ) 
(n)
 sh( x ),
=
ch( x ),
for n = 2k + 1
;
for n = 2k
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
285
ch( x ), for n = 2k + 1
=
 sh( x ), for n = 2k
are valid for the derivatives of the symmetric hyperbolic Fibonacci functions:
 sh ( x ) 
(n)
cFs ( x ) 
(n)
 sFs ( x ) 
(n)
n
 ( ln τ ) sFs( x ), for n = 2k + 1
=
n
( ln τ ) cFs( x ), for n = 2k
(5.83)
( ln τ )n cFs ( x ), for n = 2k + 1
.
=
n
ln
τ
(
),
2
sFs
x
for
n
k
=
)
(
(5.84)
Theorem 5.16 (formulas for integration). The following correlations simi
lar to the integrals of the classical hyperbolic functions
 sh( x ),
for
n = 2k + 1
∫∫∫ ch( x)dx = ch( x), for n = 2k
;
n
 ch( x ), for n = 2k + 1
=
 sh( x ), for n = 2k
n
are valid for the symmetric hyperbolic Fibonacci and Lucas functions:
∫∫∫
sh( x )dx
∫∫∫
cFs ( x )dx
 ln τ − n sFs ( x ), for n = 2k + 1
)
(
=
( ln τ )− n cFs ( x ), for n = 2k
(5.85)
∫∫∫ cLs ( x)dx
 ( ln τ ) − n sLs ( x ), for n = 2k + 1
=
−n
( ln τ ) cLs ( x ), for n = 2k
(5.86)
∫∫∫
sFs ( x )dx
 ( ln τ )− n cFs ( x ), for n = 2k + 1
=
−n
( ln τ ) sFs ( x ), for n = 2k
(5.87)
∫∫∫ sLs ( x)dx
−n
 ( ln τ ) cLs ( x ), for n = 2k + 1
=
.
−n
( ln τ ) sLs ( x ), for n = 2k
(5.88)
n
n
n
n
Alexey Stakhov
THE MATHEMATICS OF HARMONY
286
5.8. The Golden Shofar
5.8.1. The Quasisine Fibonacci and Lucas Functions
Consider Binet formulas for Fibonacci and Lucas numbers represented in
the following form:
n
τn − ( −1) τ− n
Fn =
(5.89)
5
Ln = τn + (−1)n τ− n ,
(
(5.90)
)
where τ = 1 + 5 2 is the golden mean and n= 0, ±1, ±2, ±3, … .
By comparing Binet formula (5.89) and (5.90) with the symmetric hyper
bolic Fibonacci and Lucas functions (5.53)(5.56), we can see that the continu
ous functions τx and τx in the formulas (5.53)(5.56) correspond to the discrete
sequences τn and τn in formulas (5.89) and (5.90). Then we set up in correspon
dence to the alternating sequence (1)n in the Binet formulas (5.89) and (5.90)
some continuous function, taking the values 1 and 1 at the discrete points x= 0,
±1, ±2, ±3, …. The trigonometric function cos(πx) is the simplest of them. This
reasoning is the basis for introducing the new continuous function that is con
nected with Fibonacci and Lucas numbers.
Definition 5.1. The following continuous function is called the Quasisine
Fibonacci Function (QSFF):
τ x − cos ( πx ) τ− x
.
QF ( x ) =
(5.91)
5
There is the following correlation between Fibonacci numbers Fn given by
(5.89) and the quasisine Fibonacci function given by (5.91):
τn − cos ( πn ) τ − n
Fn = Q F ( n ) =
,
(5.92)
5
where n= 0, ±1, ±2, ±3, … .
Definition 5.2. The following continuous function is called the Quasisine
Lucas Function (QSLF):
QL ( x ) = τ x + cos ( πx ) τ− x .
(5.93)
The graph of the QSFF is a quasisine curve that passes through all points
corresponding to the Fibonacci numbers that are given by (5.89) on the coordi
nate plane (Fig. 5.5). The symmetric hyperbolic Fibonacci functions (5.53) and
(5.54) (Fig. 5.3) are the envelopes of the quasisine Fibonacci function Q(x).
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
287
The graph of the QSLF is a quasisine curve that passes through all points
corresponding to the Lucas numbers that are given by (5.90) on the
coordinate plane (Fig. 5.6). The symmetric hyperbolic Lucas functions (5.55)
and (5.56) (Fig. 5.4) are the envelopes of the quasisine Lucas function QL(x).
Y
Y
y = cLs(x)
y = cFs(x)
1
1
0 1
y = FF(x)
y = sFs(x)
Figure 5.5. A graph of the
quasisine Fibonacci function
0
1
y = FL(x)
y = sLs(x)
Figure 5.6. A graph of the
quasisine Lucas function
5.8.2. Recursive Properties of the Quasisine Fibonacci and Lucas
Functions
It is easy to prove the following theorems for the quasisine Fibonacci and
Lucas functions.
Theorem 5.17. For the quasisine Fibonacci function there is the following
correlation similar to the recursive relation for the Fibonacci numbers
F n+2=Fn+1+F n:
Q F ( x + 2 ) = Q F ( x + 1) + Q F ( x ) .
(5.94)
Theorem 5.18. For the quasisine Lucas function there is the following cor
relation similar to the recursive relation for the Lucas numbers Ln+2=Ln+1+Ln:
QL ( x + 2 ) = QL ( x + 1) + QL ( x ) .
(5.95)
Theorem 5.19. For the quasisine Fibonacci function there is the following
n +1
correlation similar to Cassini’s formula Fn2 − Fn +1Fn −1 = ( −1) :
QF ( x )  − QF ( x + 1)QF ( x − 1) = − cos ( πx ) .
2
(5.96)
By analogy with Theorems 5.17 5.19 we can prove other identities for
the quasisine Fibonacci and Lucas functions (Table 5.4).
THE MATHEMATICS OF HARMONY
Alexey Stakhov
288
Table 5.4. The identities for the quasisine Fibonacci and Lucas functions
Fn + 2 = Fn +1 + Fn
Q F ( x + 2 ) = Q F ( x + 1) + Q F ( x )
Fn + 3 + Fn = 2 Fn + 2
Q F ( x + 3 ) + Q F ( x ) = 2Q F ( x + 2 )
Fn + 3 − Fn = 2 Fn +1
Q F ( x + 3 ) − Q F ( x ) = 2Q F ( x + 1)
Fn + 6 − Fn = 4 Fn + 3
Q F ( x + 6 ) − Q F ( x ) = 4Q F ( x + 3 )
Fn2 − Fn +1 Fn −1 = ( −1)
F2 n +1 = F
2
n +1
n +1
F
F
2
2
+F
L
Ln + 2 = Ln +1 + Ln
L
F
Ln + 3 + Ln = 2 Ln + 2
Ln −1 + Ln +1 = 5 Fn
Ln + 5 Fn = 2Ln +1
L
L
L
L
F
L
F
F
F
n
L
F
L
Fn + 3 − 2 Fn = Ln
Ln +1 Ln −1 − L = −5 ( −1)
2
F
2
n
2
n
[Q ( x )] − Q ( x + 1) Q ( x − 1) = − cos ( πx )
Q ( 2 x + 1) = [Q ( x + 1)] + [Q ( x )]
Q ( x + 2 ) = Q ( x + 1) + Q ( x )
Q ( x + 3 ) + Q ( x ) = 2Q ( x + 2 )
Q ( x − 1) + Q ( x + 1) = 5Q ( x )
Q ( x ) + 5Q ( x ) = 2Q ( x + 1)
Q ( x + 3 ) − 2Q ( x ) = Q ( x )
Q ( x + 1) Q ( x − 1) − [Q ( x )] = −5 cos ( πx )
L
F
L
2
L
L
L
5.8.3. Threedimensional Fibonacci Spiral
It is well known that the trigonometric sine and cosine can be defined as a
horizontal projection of the translational movement of a point on the surface of an
infinite rotating cylinder with radius 1 and the symmetry center coinciding with
the axis ОХ. Such threedimensional spiral is described by the complex function
f(x)=cos(x)+isin(x), where i = −1. The sine function is its projection on a plane.
If we assume that the quasisine Fibonacci function (5.91) is a projection of
the threedimensional spiral on some funnelshaped surface, we can then define
the socalled Threedimensional Fibonacci Spiral.
Definition 5.3. The following function is called the Threedimensional Fi
bonacci Spiral:
sin ( πx )  τ− x
.
+i 
(5.97)
5
5
This function, by its shape, reminds one of a spiral that is drawn on a well
with the end bent (Fig. 5.7).
It is easy to prove the following theorem for the threedimensional
Fibonacci spiral.
Theorem 5.20. For the threedimensional Fibonacci spiral the following cor
relation, similar to the recursive relation for Fibonacci numbers Fn+2=Fn+1+Fn,
is valid:
SF ( x ) =
τ x − cos ( πx )  τ− x
S F ( x + 2 ) = S F ( x + 1) + S F ( x ) .
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
289
Y
Z
0
X
Figure 5.7. The threedimensional Fibonacci spiral
5.8.4. The Golden Shofar
We can separate the real and imaginary parts of the threedimensional Fi
bonacci spiral (5.97):
Re  S F ( x )  =
τ x − cos ( πx )  τ− x
5
(5.98)
sin ( πx )  τ− x
Im  S F ( x )  =
.
(5.99)
5
The following system of equations can be obtained from (5.97), (5.98), and
(5.99) if we consider the axis OY as a real axis and the axis OZ as an imaginary axis:

cos ( πx )  τ− x
τx
=−
y ( x ) −

5
5

−x
sin ( πx )  τ

z ( x ) =
5

(5.100)
Let us square both expressions of the equation system (5.100) and add them
together. Taking y and z as independent variables, we obtain a curvilinear sur
face of the second degree called the Golden Shofar.
THE MATHEMATICS OF HARMONY
Alexey Stakhov
290
Definition 5.4. The following curvilinear function of the second degree is
the Golden Shofar:
2
2
 τ− x 

τx 
2
+
=
y
−
z



 .
(5.101)
5

 5
We can see in Fig. 5.8 a threedimension
al surface corresponding to (5.101). It is sim
ilar to the horn or well with a narrow end. In
the Hebrew language the word “Shofar”
means horn which is a symbol of power.
The formula for the Golden Shofar can
be represented in the following form:
z 2 = cFs ( x ) − y  ×  sFs ( x ) + y  ,
(5.102)
Figure 5.8. The Golden Shofar
where sFs(x) and cFs(x) are the symmetric
hyperbolic Fibonacci sine and cosine, respectively.
A projection of the Golden Shofar on the plane XOY is shown in Fig. 5.9. The
Golden Shofar is a projection into the space between the graphs of the symmet
ric hyperbolic Fibonacci sine and cosine (Fig. 5.3).
Y
Z
y = cFs(x)
z = ax/√5
1
0
1
1
X
0
y = sFs(x)
1
X
z = ax/√5
Figure 5.10. A projection of the
Golden Shofar on the plane XOZ
Figure 5.9. A projection of the
Golden Shofar on the plane XOY
The function (5.97) lies on the Golden Shofar and “pierces” the plane XOY
at the points that correspond to the Fibonacci sequence (Fig. 5.9).
A projection of the Golden Shofar on the plane XOZ is shown in Fig. 5.10.
The Golden Shofar is a projection into the space between the graphs of the two
exponent functions − τ x 5 and τ − x 5 .
By cutting the Golden Shofar by the planes that are parallel to the plane
YOZ, the circles with the center 0;τ x 5 and the radius τ − x 5 are obtained.
(
)
(
(
)
)
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
291
It is possible to say that the function y ( x ) = τ x 5 is the pseudoaxis of sym
metry (or the axis of pseudosymmetry) of the Golden Shofar (Fig. 5.8).
It is possible to assume that the Golden Shofar is a new model of the field
with curvilinear structure and similar to the model of the gravitational well
used in the general theory of relativity.
5.8.5. A General Model of the Hyperbolic Space with a “Shofarable”
Topology
Based on experimental data obtained in 2003 by the NASA Wilkinson
Microwave Anisotropy Probe (WMAP), a new hypothesis about the structure
of the Universe was developed. According to [155], the geometry of the Uni
verse is similar in shape to a horn or a pipe with an extended bell. As a result of
this discovery we can make the following claim:
The Universe has a “shofarlike” topology as shown in Fig. 5.11.
Figure 5.11. The “shofarlike” topology
5.9. A General Theory of the Hyperbolic Functions
5.9.1. A Definition of the Hyperbolic Fibonacci and Lucas mFunctions
Alexey Stakhov and Boris Rozin introduced [106] a new class of hyperbol
ic functions, the Symmetric Hyperbolic Fibonacci and Lucas Functions, based
on an analogy between Binet formulas and the classical hyperbolic functions.
By using this approach, Alexey Stakhov introduced [118] the Hyperbolic Fi
bonacci and Lucas Functions of the Order m or simply Hyperbolic Fibonacci
and Lucas mFunctions. These functions are based on an analogy between Gaza
le formulas that are given by (4.281) and (4.292) with the classical hyperbolic
Alexey Stakhov
THE MATHEMATICS OF HARMONY
292
functions (5.7) and (5.8). Consider the new class of hyperbolic Fibonacci and
Lucas functions introduced in [118].
Hyperbolic Fibonacci msine
x
−x


Φ mx − Φ m− x
m + 4 + m2   m + 4 + m2  
1


 −

sFm ( x ) =
=
 
  (5.103)
2
2
4 + m2
4 + m 2 


 

Hyperbolic Fibonacci mcosine
x
−x

2 
 m + 4 + m2  
Φ mx + Φ m− x
1
4
m
+
+
m

 +
  (5.104)
cFm ( x ) =
=
2
2 


 
2
2
4+m
4 + m 
 
 

Hyperbolic Lucas msine
x
−x
 m + 4 + m2   m + 4 + m2 
x
−x
 −

sLm ( x ) = Φ m − Φ m = 
(5.105)

 

2
2

 

Hyperbolic Lucas mcosine
x
−x
 m + 4 + m2   m + 4 + m2 
x
−x
 +

cLm ( x ) = Φ m + Φ m = 
(5.106)

 

2
2

 

It is easy to prove that the Fibonacci and Lucas mnumbers are determined
identically by the hyperbolic Fibonacci and Lucas mfunctions as follows:
 sFm ( n ) for n = 2k
Fm ( n ) = 
cFm ( n ) for n = 2k + 1
(5.107)
cLm ( n ) for n = 2k
Lm ( n ) = 
 sLm ( n ) for n = 2k + 1
Graphs of the hyperbolic Fibonacci and Lucas mfunctions are similar to
graphs of the classical hyperbolic functions. Here it is important to note that
at the point x=0, the hyperbolic Fibonacci mcosine cFm(x) (5.104) takes on
the value cFm ( x ) = 2 4 + m2 , and the hyperbolic Lucas mcosine cLm(x)
takes on the value cLm(0)=2. It is also important to emphasize that the Fibonac
ci mnumbers Fm(n) with the even indices n=0,±2,±4,±6,… are “inscribed” into
the graph of the hyperbolic Fibonacci msine sFm(x) at the discrete points
x=0,±2,±4,±6,… and the Fibonacci mnumbers Fm(n) with odd indices
n=±1,±3,±5,… are “inscribed” into the hyperbolic Fibonacci mcosine cFm(x) at
the discrete points x=±1,±3,±5,…. On the other hand, the Lucas mnumbers
Lm(n) with even indices are “inscribed” into the graph of the hyperbolic Lucas
mcosine cLm(x) at the discrete points x=0,±2,±4,±6,… and the Lucas mnum
bers Lm(n) with odd indices are “inscribed” into the graph of the hyperbolic Lu
cas msine cLm(x) at the discrete points x=±1,±3,±5,… .
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
293
We can also introduce the notions of hyperbolic Fibonacci and Lucas m
tangents and mcotangents.
Hyperbolic Fibonacci mtangent
tFm ( x ) =
sFm ( x )
cFm ( x )
=
Φ mx − Φ m− x
Φ mx + Φ m− x
(5.108)
Hyperbolic Fibonacci mcotangent
cFm ( x )
Φ mx + Φ −mx
ctFm ( x ) =
=
sFm ( x ) Φ mx − Φ −mx
(5.109)
Hyperbolic Lucas mtangent
tLm ( x ) =
sLm ( x )
cLm ( x )
=
Φ mx − Φ −mx
Φ mx + Φ −mx
(5.110)
Hyperbolic Lucas mcotangent
ctLm ( x ) =
cLm ( x )
sLm ( x )
=
Φ mx + Φ m− x
Φ mx − Φ −mx
(5.111)
By analogy we can introduce other hyperbolic Fibonacci and Lucas mfunc
tions, in particular, secant, cosecant, and so on.
5.9.2. General Properties of the Hyperbolic Fibonacci and Lucas m
Functions
It is easy to prove that the function (5.103) is an odd function because
sFm ( − x ) =
Φ m− x − Φ mx
cFm ( − x ) =
Φ m− x + Φ mx
=−
4 + m2
On the other hand,
Φ mx − Φ m− x
4 + m2
= − sFm ( x ) .
(5.112)
Φ mx + Φ m− x
= cFm ( x ) ,
(5.113)
4 + m2
4 + m2
that is, the hyperbolic Fibonacci mcosine (5.104) is an even function.
By analogy, we can prove that the hyperbolic Lucas msine (5.105) is an
odd function and the hyperbolic Lucas mcosine (5.106) is an even function,
that is,
sLm ( x ) = − sLm ( − x )
=
(5.114)
cLm ( x ) = cLm ( − x ) .
(5.115)
The properties (5.112)(5.15) show that the symmetric hyperbolic Fibonac
ci and Lucas mfunctions (5.103)(5.106) possess the property of evenness (5.52).
THE MATHEMATICS OF HARMONY
Alexey Stakhov
294
By making the pairwise comparison of the functions (5.108) with (5.110),
and (5.109) with (5.111), we can conclude that the hyperbolic Fibonacci and
Lucas mtangents and mcotangents are coincident, respectively, that is, we have:
tFm ( x ) = tLm ( x ) and ctFm ( x ) = ctLm ( x ) .
(5.116)
It is easy to prove that the functions (5.108) and (5.109) are odd func
tions because
Φ − x − Φ mx
tFm ( − x ) = −mx
= −tFm ( x )
Φ m + Φ mx
ctFm ( − x ) =
Φ m− x + Φ mx
= −ctFm ( x ) .
Φ m− x − Φ mx
Taking into consideration (5.116) we can write:
tLm ( − x ) = −tLm ( x )
ctLm ( − x ) = −ctLm ( x ) ,
that is, the functions (5.110) and (5.111) are also odd functions.
5.9.3. Partial Cases of the Hyperbolic Fibonacci and Lucas mFunctions
Consider the partial cases of the hyperbolic Fibonacci and Lucas mfunc
tions (5.103)(5.106) for different values of the order m.
Hyperbolic Fibonacci and Lucas 1functions
x
−x
Φ1x − Φ1− x
1  1 + 5   1 + 5  
sF1 ( x ) =
=

 −

(5.117)
5
5  2   2  


x
−x
1  1 + 5   1 + 5  
Φ1x + Φ1− x
cF1 ( x ) =
=

 +

(5.118)
5
5  2   2  


x
−x
 1+ 5   1+ 5 
x
−x
−
sL1 ( x ) = Φ1 − Φ1 = 
(5.119)
 2   2 

 

−x
x
5  1+ 5 
cL1 ( x )

 +
 2   2 
Hyperbolic Fibonacci and Lucas 2functions
1+
= Φ1x + Φ1− x = 
sF2 ( x ) =
cF2 ( x ) =
Φ 2x − Φ 2− x
8
Φ 2x + Φ 2− x
8
(5.120)
(
) (
)
(5.121)
(
) (
)
(5.122)
=
x
−x
1 

1
2
1
2
+
−
+


2 2
=
x
−x
1 
1 + 2 + 1 + 2 


2 2
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
295
( ) − (1 + 2 )
cL ( x ) = Φ + Φ = (1 + 2 ) + (1 + 2 )
sL2 ( x ) = Φ 2x − Φ 2− x = 1 + 2
x
2
2
−x
2
x
−x
x
−x
(5.123)
(5.124)
Hyperbolic Fibonacci and Lucas 3functions
x
−x
Φ 3x − Φ 3− x
1  3 + 13   3 + 13  
sF3 ( x ) =
=

 −

2  
2  
13
13 


x
−x

x
−x
Φ + Φ3
1  3 + 13   3 + 13  
cF3 ( x ) = 3
=

 +

2  
2  
13
13 


sL3 ( x )

3+
= Φ 3x − Φ 3− x = 


x
13   3 + 13 

 −
2   2 
x
(5.125)
(5.126)
−x
(5.127)
−x
13   3 + 13 
cL3 ( x )

 +
(5.128)
2  
2 

Note that a list of these functions can be continued ad infinitum.
It is easy to see that the functions (5.103)(5.106) are connected by very
simple correlations:

3+
= Φ 3x + Φ 3− x = 

sFm ( x ) =
sLm ( x )
; cFm ( x ) =
cLm ( x )
.
(5.129)
4 + m2
4 + m2
This means that the hyperbolic Lucas mfunctions (5.103) and (5.104) coin
cide with the hyperbolic Fibonacci mfunctions (5.105) and (5.106) to within
the constant coefficient 1 1 + m2 .
5.9.4. Comparison of the Classical Hyperbolic Functions with the
Hyperbolic Lucas mFunctions
Let us compare the hyperbolic Lucas mfunctions (5.105) and (5.106) with
the classical hyperbolic functions (5.7) and (5.8). For the case
4 + m2 + m
=e
(5.130)
2
the hyperbolic Lucas mfunctions (5.105) and (5.106) coincide with the classical
hyperbolic functions (5.7) and (5.8) to within the constant coefficient 1/2, that is,
sL ( x )
cLm ( x )
sh ( x ) = m
.
and ch ( x ) =
(5.131)
2
2
By using (5.130), after simple transformations we can calculate the value
Φm =
Alexey Stakhov
THE MATHEMATICS OF HARMONY
296
me, for which the equality (5.130) is valid:
1
me = e − ≈ 2.35040238...
(5.132)
e
Thus, according to (5.131) the classical hyperbolic functions (5.7) and (5.8)
are a partial case of the hyperbolic Lucas mfunctions, if m is equal to (5.132).
As the classical hyperbolic functions (5.7) and (5.8) are a partial case of
(5.105) and (5.106), we have the right to assert that the formulas (5.103) through
(5.106) represent a general class of hyperbolic functions.
5.9.5. Recursive Properties of the Hyperbolic Fibonacci and Lucas m
Functions
The hyperbolic Fibonacci and Lucas mfunctions possess recursive prop
erties similar to Fibonacci and Lucas mnumbers that are given by the recur
sive relations (4.250) and (4.302). On the other hand, they possess all of the
hyperbolic properties similar to the properties of the classical hyperbolic
functions. Let us first prove the recursive properties for the hyperbolic Fi
bonacci and Lucas mfunctions.
Theorem 5.21. The following correlations that are similar to the recursive
relation for the Fibonacci mnumbers Fm(n+2)=mFm(n+1)+Fm(n) are valid
for the hyperbolic Fibonacci mfunctions:
sFm ( x + 2 ) = mcFm ( x + 1) + sFm ( x )
(5.133)
cFm ( x + 2 ) = msFm ( x + 1) + cFm ( x ) .
Proof:
Φ x +1 + Φ m− x −1 Φ mx − Φ m− x
mcFm ( x + 1) + sFm ( x ) = m m
+
4 + m2
4 + m2
(5.134)
=
Φ mx ( mΦ m + 1) − Φ m− x (1 − mΦ m−1 )
.
(5.135)
4 + m2
As mΦ m + 1 = Φ 2m and 1 − Φ m−1 = Φ m−2 , we can represent (5.135) as follows:
Φ x + 2 − Φ −mx −2
mcFm ( x + 1) + sFm ( x ) = m
= sFm ( x + 2 )
4 + m2
that proves the identity (5.133).
By analogy, we can prove the identity (5.134).
Theorem 5.22 (a generalization of Cassini formula).
The following correlations, that are similar to the Cassini formula
n +1
2
Fm ( n ) − Fm ( n − 1) Fm ( n + 1) = ( −1) , are valid for the hyperbolic Fibonacci
mfunctions:
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
297
 sFm ( x )  − cFm ( x + 1) cFm ( x − 1) = −1
2
cFm ( x )  − sFm ( x + 1) sFm ( x − 1) = 1.
Proof:
2
(5.136)
(5.137)
 sFm ( x ) − cFm ( x + 1) cFm ( x − 1)
2
2
 Φ x − Φ m− x  Φ mx+1 + Φ m− x −1 Φ mx−1 + Φ m− x +1
= m
×
 −
4 + m2
4 + m2
 4 + m2 
=
=
Φ 2mx − 2 + Φ m−2 x − ( Φ 2mx + Φ 2m + Φ m−2 + Φ m−2 x )
−2 − ( Φ 2m + Φ m−2 )
4 + m2
(5.138)
.
4 + m2
By using the Binet formula (4.292), for the case n=2 we can write:
Lm ( 2 ) = Φ 2m + Φ m−2 .
(5.139)
By using the recursive formula (4.290) and the seeds (4.287) and (4.288),
we can represent the Lucas mnumber Lm(2) as follows:
(5.140)
Lm ( 2 ) = mLm (1) + Lm ( 0 ) = m × m + 2 = m 2 + 2.
Taking into consideration (5.140), we can conclude from (5.138) that the
identity (5.136) is valid.
By analogy, we can prove the identity (5.137).
5.9.6. Hyperbolic Properties of the Hyperbolic Fibonacci and Lucas m
Functions
Theorem 5.23. The following identity, that is similar to the identity [ch(x)]2
[sh(x)]2=1 for the classical hyperbolic functions, is valid for the hyperbolic Fi
bonacci mfunctions:
2
2
4
cFm ( x )  −  sFm ( x )  =
.
(5.141)
4 + m2
Proof:
2
2
2
2
 Φ mx + Φ m− x   Φ mx − Φ m− x 
cFm ( x ) −  sFm ( x ) = 

 −
 4 + m2   4 + m2 
Φ 2 x + 2 + Φ m−2 x − Φ 2mx + 2 − Φ m−2 x
4
= m
.
=
2
4+m
4 + m2
Theorem 5.24. The following identity, that is similar to the identity
[ch(x)]2 [sh(x)]2=1 for the classical hyperbolic functions, is valid for the hyper
bolic Lucas mfunctions:
Alexey Stakhov
THE MATHEMATICS OF HARMONY
298
2
2
(5.142)
cLm ( x )  −  sLm ( x )  = 4.
The proof is analogous to Theorem 5.23.
Theorem 5.25. The following identity, that is similar to the identity
ch(x+y)=ch(x)ch(y)+sh(x)sh(y) for the classical hyperbolic functions, is valid
for the hyperbolic Fibonacci mfunctions:
2
cFm ( x + y ) = cFm ( x ) cFm ( y ) + sFm ( x ) sFm ( y ) .
(5.143)
4 + m2
Proof:
cFm ( x ) cFm ( y ) + sFm ( x ) sFm ( y )
=
Φ mx + Φ m− x
×
Φ my + Φ m− y
+
Φ mx − Φ m− x
×
Φ my − Φ m− y
4 + m2
4 + m2
4 + m2
4 + m2
Φ x + y + Φ mx− y + Φ m− x + y + Φ m− x − y + Φ mx+ y − Φ mx− y − Φ m− x + y + Φ m− x − y
= m
4 + m2
2 ( Φ mx+ y + Φ m− x − y )
2
cFm ( x + y ) .
=
=
4 + m2 × 4 + m2
4 + m2
Theorem 5.26. The following identity, that is similar to the identity ch(x
y)=ch(x)ch(y)sh(x)sh(y) for the classical hyperbolic functions, is valid for
the hyperbolic Fibonacci mfunctions:
2
cFm ( x − y ) = cFm ( x ) cFm ( y ) − sFm ( x ) sFm ( y ) .
(5.144)
4 + m2
The proof is analogous to Theorem 5.25.
By analogy, we can prove the following theorems for the hyperbolic Fibonacci
and Lucas mfunctions.
Theorem 5.27. The following identities, that are similar to the identity
ch(2x)=[ch(x)]2+[sh(x)]2 for the classical hyperbolic functions, are valid for the
hyperbolic Fibonacci and Lucas mfunctions:
2
2
2
cFm ( 2 x ) = cFm ( x )  +  sFm ( x )
(5.145)
5
2cLm ( 2 x ) = cLm ( x )  +  sLm ( x )  .
2
2
(5.146)
Theorem 5.28. The following identities, that are similar to the identity
sh(2x)=2sh(x)ch(x) for the classical hyperbolic functions, are valid for the hy
perbolic Fibonacci and Lucas mfunctions:
1
sFm ( 2 x ) = sFm ( x ) cFm ( x )
(5.147)
4 + m2
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
299
sLm ( 2 x ) = sLm ( x ) cLm ( x ) .
(5.148)
Theorem 5.29. The following formulas, that are similar to Moivre’s formu
las [ch(x)±sh(x)]n =ch(nx)±sh(nx) for the classical hyperbolic functions, are
valid for the hyperbolic Fibonacci and Lucas mfunctions:
n −1


n
2
cFm ( x ) ± sFm ( x )  = 
  cFm ( nx ) ± sFm ( nx ) 
(5.149)

2 
 4+m 
(5.150)
cLm ( x ) ± sLm ( x ) = 2n −1 cFm ( nx ) ± sFm ( nx )  .
Thus, our research results in a general theory of hyperbolic functions based on
the Gazale formulas (4.281) and (4.291). Over several centuries, science, in partic
ular, mathematics and theoretical physics, made wide use of classical hyperbolic
functions with base e. These functions were used by Lobachevsky in his nonEu
clidean geometry and Minkowski in his geometric interpretation of Einstein’s rel
ativity theory. More recently Ukrainian mathematicians Stakhov, Tkachenko and
Rozin [51, 98, 106, 116, 118, 119] broke monopoly on classical hyperbolic func
tions in contemporary mathematics and theoretical physics. It is now clear that
the above hyperbolic Fibonacci and Lucas mfunctions based on the Gazale for
mulas infinitely extendable to new hyperbolic models of Nature. It is difficult to
imagine that the set of new hyperbolic functions has the same cardinality as the
set of real numbers because every positive real number m generates its own kind of
hyperbolic functions! And all of them possess unique recursive and hyperbolic
properties similar to the properties of classical hyperbolic functions and the sym
metric hyperbolic Fibonacci and Lucas functions introduced in [106, 116, 119].
n
5.10. A Puzzle of Phyllotaxis
5.10.1. The Phenomenon of Phyllotaxis
In Chapter 2 we described the botanical phenomenon known as Phyllotaxis.
This phenomenon is inherent in many biological objects. On the surface of the
bioorgans, these objects (sprouts of plants and trees, seeds on the disks of sun
flower heads and pine cones, etc.) are arranged in clockwise and counterclock
wise spirals. There are two types of phyllotaxis. The first type concerns the dispo
sition of branches of plants and trees, and the second type concerns the dense
packed phyllotaxis objects such as that of a pinecone, pineapple, or the disk of a
sunflower head. In the first case, the regularities of phyllotaxis are described by
the ratios of adjacent Fibonacci numbers taken through one, that is,
Alexey Stakhov
THE MATHEMATICS OF HARMONY
300
(5.151)
cLm ( x ) ± sLm ( x ) = 2n −1 cFm ( nx ) ± sFm ( nx )  .
Each plant and tree possesses its own characteristic ratio taken from
(5.151). This ratio is named the Phyllotaxis Order of a given plant or tree.
The phyllotaxis orders are different for different plants and trees, for exam
ple, for linden, elm, beech and cereals the phyllotaxis order is equal to (2/1);
for alder, hazel and grape (3/1); for oak and cherry (5/2); for raspberry, pear,
poplar and barberries (8/3); and for almonds (13/5).
Note that trees and plants are subject primarily to the laws of Fibonacci
phyllotaxis (5.151); in rare cases the plants are subject to Lucas phyllotaxis
(Lucas numbers 2, 1, 3, 4, 7, 11, 18, 29, …) or in a few cases phyllotaxis based on
the numerical sequence: 5, 2, 7, 9, 16, 25, … that also satisfies the Fibonacci
recursive formula.
The other type of phyllotaxis is represented in the form of denselypacked
botanical objects such as the head of a sunflower, pinecone, pineapple or cactus,
etc. For such phyllotaxis objects, it is used usually the number ratios of the left
hand and righthand spirals observed on the surface of the phyllotaxis objects.
These ratios are equal to the ratios of adjacent Fibonacci numbers, that is,
Fn +1 2 3 5 8 13 21
1+ 5
:
, , , , , ,... → τ =
.
(5.152)
Fn
1 2 3 5 8 13
2
For example, the head of a sunflower can have the phyllotaxis orders given
by Fibonacci ratios: 89/55, 144/89 and even 233/144.
n
5.10.2. Dynamic Symmetry
When observing phyllotaxis the question arises: how do Fibonacci spirals
forming on the surface during growth? This problem is one of the most intrigu
ing Puzzles of Phyllotaxis. Its essence consists in the fact that the majority of
bioforms change their phyllotaxis orders during their growth. It is known, for
example, that sunflower disks that are located on different levels of the same
stalk have different phyllotaxis orders; moreover, the greater the age of the
disk, the higher its phyllotaxis order tends to be. This means that during the
growth of the phyllotaxis object, a natural modification (or increase) of the
symmetry relation occurs, and this modification of symmetry obeys the law:
2 3
5 8 13
21
→ → → →
→
→ ....
(5.153)
1 2
3 5
8
13
The modification of the phyllotaxis orders according to (5.153) is called
Dynamic Symmetry [37]. Many scientists who study this puzzle of phyllotax
is believe that the phenomenon of dynamic symmetry (5.153) is of fundamen
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
301
tal interdisciplinary importance. Recall that, in Vernadsky’s opinion, the prob
lem of biological symmetry is the key problem of biology.
Thus, the phenomenon of dynamic symmetry (5.153) plays a special role in
the geometric problem of phyllotaxis. One may assume that this numerical reg
ularity (5.153) reflects some general geometric laws that hide the secret of the
dynamic mechanism of phyllotaxis, and uncovering it would be of great impor
tance for understanding the phyllotaxis phenomenon in general.
A new geometric theory of phyllotaxis was developed recently by the Ukrai
nian architect Oleg Bodnar. This original theory is stated in Bodnar’s books [37,
52]. However, in order to understand Bodnar’s research more fully, we have to
get a better understanding of the geometric theory of hyperbolic functions [156].
5.11. A Geometric Theory of the Hyperbolic Functions
5.11.1. Compression and Expansion
We start the study of geometric theory of hyperbolic functions from the impor
tant geometric transformation of the hyperbolic geometry. This is a geometric trans
formation named the Compression to a Straight Line о with the Compression Coeffi
cient k (Fig. 5.12). This transformation consists of the following: Every point А of the
plane passes into the point А′ that lies on the ray РА perpendicular to о, here the
ratio PA′ : PA=k or PA′ = kPA (Fig. 5.12a). If the compression coefficient k>1, then
PA′ > PA (Fig. 5.12b); in this case the transformation could be named an Expansion
from a Straight Line о. It is clear that the expansion using the coefficient k is equiva
lent to the compression with coefficient (1/k).
A compression and an expansion possess
A′
a number of important properties [156]:
A
1. At the compression (expansion) every
A
straight line passes on into a straight line.
A′
2. At the compression (expansion) all
parallels pass on into parallels.
3. At the compression (expansion) the ra
tio of the segments lying on one straight
o
P
o
P
line remain constant.
(a)
(b)
4. At the compression (expansion), the
Figure 5.12. The compression of a
point A to a straight line (a) and the areas of all figures change in a constant ra
expansion from a straight line (b)
tio equal to the compression coefficient k.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
302
5.11.2. Hyperbola
Consider the important geometric curve called a Hyperbola. It is described
by the following equality:
y=a/x or xy=a.
A graph of the hyperbola is represented in Fig. 5.13.
(5.154)
y
b
Q
a
M
Q
M
M1
P1
O2
O
P
O
M2
a
Figure 5.13. Hyperbola
P
x
Q1
P2
b
Figure 5.14. The axes of a hyperbola
It follows from (5.154) and Fig. 5.13 that a graph of the hyperbola consists of
two branches that are located for the case a>0 in the first quadrant (x and y are
positive) and in the third quadrant (x and y are negative) of the coordinate sys
tem. Geometrically, the branches of the hyperbola aim towards the coordinate
axes, but they never intersect themselves. This means that the coordinate axes are
asymptotes of the hyperbola. Note that the equation xy=a has a simple geometric
interpretation: the area of the rectangles MQOP or M′Q′O′P′ that are bounded by
the coordinate axes and the straight lines that are drawn through any points M
and M′ of the hyperbola in parallel to the coordinate axes (Fig. 5.13) is equal to a,
that is, this area does not depend on the choice of the points M and M′. We will
name these rectangles MQOP and M′Q′O′P′ with area equal to a the Coordinate
Rectangles of the points M and M′. Then we can give the following geometric def
inition of the hyperbola [156]:
“A hyperbola is the geometric location of the points that lie in the first and
third quadrants of the coordinate system with coordinate rectangles that
have constant area.”
It is easy to prove that the origin of coordinate O is the Centre of Symmetry of
the hyperbola, that is, the branches of the hyperbola are symmetric one to other
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
303
with respect to the origin of coordinates O. The hyperbola also has the Axes of Sym
metry, the bisectors of the coordinate angles аа and bb (Fig. 5.14). The centre of
symmetry O and the axes of symmetry аа and bb are frequently called simply the
Center and the Axes of the hyperbola; the points A and B, in which the hyperbola is
intersected with the axis аа, are called Tops of the hyperbola.
Hereinafter, we will use analogies between hyperbola and circle. With this
purpose in mind we will introduce, first of all, the concept of the Diameter of a
hyperbola; every line segment passing through the centre of the hyperbola and
connecting the points of the opposite branches of the hyperbola is called a
Diameter of the Hyperbola (it is similar to the diameter of a circle passing through
its centre). Let us also introduce the concept of a radius of the hyperbola; a line
segment, going from the centre of the hyperbola up to the crossing point with
the hyperbola, is called a radius of the hyperbola (that is, the radii of the hyper
bola are determined similarly to the radii of the circle).
5.11.3. Geometric Definition of the Hyperbolic Functions
The hyperbola in Fig. 5.13 and Fig. 5.14 is the basis for a geometric
definition of hyperbolic functions. The geometric theory of Hyperbolic Func
tions or Hyperbolic Trigonometric Functions is similar to the theory of tradi
tional Circular Trigonometric Functions. In order to emphasize an analogy
between hyperbolic and circular trigonometric functions, we will state the
theory of the hyperbolic functions in parallel with the theory of circular
trigonometric functions. We can choose the axis of symmetry of the hyper
bola by the coordinate axes as is shown in Fig. 5.15 and then we use this
geometric representation of the hyperbola for the geometric definition of
the hyperbolic functions.
Y
Y
M
N
M
N
X
B
O
P1 P
A X
B
O
P
A
P1
N1
M1
(a)
M1
N1
(b)
Figure 5.15. The unit circle (a) and the unit hyperbola (b)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
304
Let us examine the unit circle (Fig. 5.15a). We can see in Fig. 5.15a a Circular
Sector OMA bounded by the radii OM, OA and the arc MA. The number that is
equal to the length of the arc АМ or equal to the double area of the sector, bound
ed by the radii OM and OA and the arc MA, is called a Radian Angle α between the
radii ОА and ОМ of the circle. We can now drop the perpendicular МР to the
diameter ОА from the point М of the circle; at the point A we draw a tangent to
the circle up to its intersection with the diameter ОМ at the point N. The line
segment РМ of the perpendicular is called a Line of Sine, the line segment ОР of
the diameter is called a Line of Cosine and the line segment AN is called a Line of
Tangent. The lengths of the line segments РМ, ОР and AN are equal respectively
to the Sine, Cosine and Tangent of the angle α, that is,
y=
y=
sh
x
y = thx
x
shx
y = thx
y=
PM=sht, OP=cht, AN=tht.
As is wellknown, the circular trigono
metric functions are changing periodical
ly with the period 2π. In contrast with this
the hyperbolic functions are not periodic.
It follows from Fig. 5.15b that the hyper
bolic angle t changes from 0 up to ∞. It fol
lows from the definition of hyperbolic
functions that at the change of the hyper
bolic angle from 0 up to ∞, the hyperbolic
sine sht is changing from 0 up to ∞, the hy
perbolic cosine cht is changing from 1 up
ch
x
PM=sinα, OP=cosα, AN=tgα.
Now, let us examine the unit hyperbola (Fig. 5.15b) X2 – Y2 =1. We can see
in Fig.20b the Hyperbolic Sector OMA bounded by the hyperbolic radii OM,
OA and the hyperbolic arc MA. Then, the number that is equal to the double
area of the hyperbolic sector, bounded by these radii OM, OA and the arc MA of
the hyperbola, is called the Hyperbolic Angle t between the hyperbolic radii ОА
and ОМ. We can now drop the perpendicular МР from the point М of the hyper
bola to the diameter ОА, which is a symmetry axis, intersecting the hyperbola at
the top A. Next we draw a tangent to the hyperbola to its intersection with the
radius ОМ at point N. The line segment РМ of the perpendicular is called a Line
of Hyperbolic Sine, the line segment ОР of the axis X is called a Line of Hyper
bolic Cosine and the line segment AN is called a Line of Hyperbolic Tangent. The
lengths of the line segments РМ, ОР and AN are equal respectively to the Hyper
bolic Sine, Hyperbolic Cosine and Hyper
y
bolic Tangent of the hyperbolic angle t,
that is,
Figure 5.16. Graphs of the hyperbolic
functions
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
305
to ∞ and the hyperbolic tangent tht is changing from 0 up to 1. The graphs of
these functions are represented in Fig. 5.16.
Based upon this geometric approach, it is easy to obtain the basic rela
tions for the circular and hyperbolic trigonometric functions.
It follows from the similarity of the triangles ОМР and ОNА (Fig. 5.15a) that
AN PM
=
.
OA OP
However, AN/OA=tgα (because OA=1), and PM/OP=sinα/cosα. Thus, we
have: tgα= sinα/cosα.
Further, the coordinates of any point М of the circle are equal OP=X,
PM=Y. However, then the following important identity follows from the
unit circle equation X2+Y2=1:
OP2+PM2 = 1
or
cos2α+sin2α=1.
By dividing both parts of the obtained identity at first by cos2α and then by
sin2α we get the following remarkable formulas for the trigonometric functions:
1+tg2α=1/cos2α
ctg2α+1=1/sin2α.
It follows from a similarity of the triangles ОМР and ОNА (Fig. 5.15b) that
AN PM
=
.
OA OP
However, AN/OA=tgt (because OA=1), and PM/OP=sht/cht. Thus, we
have: tgt= sht/cht.
Further, the coordinates of any point М of the hyperbola are equal
OP = X , PM = Y . However, then the following important identity follows
from the unit hyperbola equation
X2Y2=1:
OP2PM2 = 1
or
ch2t – sh2t=1.
By dividing both parts of the obtained identity at first by ch2t and then by
sh2t we get the following remarkable formulas for the hyperbolic functions:
1th2t=1/ch2t
cth2t1=1/sh2t.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
306
By using the geometric approach, we can prove many other identities for
the trigonometric and hyperbolic functions.
5.11.4. Hyperbolic Rotation
Next we will study the hyperbola xy=a. First we make the compres
sion of a plane to the axis x with the compression coefficient k. In this case
the hyperbola xy=a passes on into the hyperbola xy=ka because the ab
scissa x remains without change and the ordinate y is replaced by yk. Then,
we make one more compression of a plane to the axis y with coefficient 1/k.
Note that the compression with coefficient 1/k is equivalent to the expan
sion with coefficient k. After the fulfilment of the compression to axis y
with coefficient 1/k it is equivalent to the extension from axis y with the
same coefficient k, the hyperbola xy=ka passes on into the hyperbola
xy=(ka/k)=a, because the ordinate y of each point for the case of new com
pression to the axis y does not vary, and the abscissa x passes on into x/k.
Thus, we can see that the sequential compression of the plane to the axis x
with the compression coefficient k and then to the axis y with the compres
sion coefficient 1/k transforms the hyperbola xy=a into itself. A sequence
of these two compressions of the plane to a straight line represents the im
portant geometric transformation called Hyperbolic Rotation. The title of
the hyperbolic rotation reflects the fact that in such transformation all
points of the hyperbola act as though they “glide on a curve,” that is, the
hyperbola acts as though it “rotates.”
Once again, note that the hyperbolic rotation is the sequential fulfilment of
the two geometric transformations, at first the compression of a plane with the
coefficient k to the axis x and then the expansion of a plane from the axis y with
the same coefficient k.
From the above properties of the “compression” and “extension,” the follow
ing properties of the hyperbolic rotation result:
1. At the hyperbolic rotation every straight line passes on into a straight
line.
2. At the hyperbolic rotation the coordinate axes (the asymptotes of the
hyperbola) pass on into themselves.
3. At the hyperbolic rotation parallels pass on into parallels.
4. At the hyperbolic rotation the ratios of all segments, lying on one and
the same straight line, remain constant.
5. At the hyperbolic rotation, the areas of all transferred figures remain
constant.
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
307
It is very important to emphasize that by means of the choice of the appro
priate value of the coefficient k, and by means of the hyperbolic rotation we can
transfer each point of the hyperbola into any other point of the same hyperbola.
In fact, the compression to the axis x with a given coefficient k transfers the point
(x,y) of the hyperbola xy=a into the point (x,ky) of the other hyperbola xy=ka;
after that the extension of the point (x,ky) from the axis y with the same coeffi
cient k transfers the point (x,ky) of the hyperbola xy=ka into the point (x/y,ky)
of the initial hyperbola. Thus, a result of the hyperbolic rotation the point (x,y)
passes on into the point (x/y,ky) of the initial hyperbola. It follows from here
that by means of a suitable hyperbolic rotation we can transfer the point (x,y) of
the hyperbola into the point (x1,y1) of the same hyperbola provided we take the
compression coefficient k=x/x1.
5.12. Bodnar’s Geometry
5.12.1. Structural numerical Analysis of Phyllotaxis Lattices
As previously noted, the Ukrainian architect Oleg Bodnar recently made an
attempt to uncover the Puzzle of Phyllotaxis with a new phyllotaxis geometry
(“Bodnar’s geometry”) presented in his books [37, 52].
To understand Bodnar’s ge
ometry let us study a cedar cone
as a characteristic example of a
phyllotaxis object (Fig. 5.17a).
On the surface of the cedar
cone every seed is blocked by
the adjacent seeds in three di
a
b
rections. As a result, we can see
the picture consisting of three
types of spirals equal to the Fi
bonacci numbers: 3, 5 and 8. In
an effort to simplify the geo
metric model of the phyllotax
is object in a Fig. 5.17a, b, we
d
c
can represent the phyllotaxis
Figure 5.17. Analysis of structurenumerical
object in a cylindrical form (Fig.
properties of the phyllotaxis lattice
Alexey Stakhov
THE MATHEMATICS OF HARMONY
308
5.17c). If we cut the surface of the cylinder in Fig. 5.17c by the vertical straight
line and then unroll the cylinder on a plane (Fig. 5.17d), we obtain a fragment of
the phyllotaxis lattice that is bounded by two parallel straight lines that are
traces of the cutting line. We can see that the three groups of parallel straight
lines in Fig. 5.17d, namely, the three straight lines 021, 116, 28 with the right
hand small declination; the five straight lines 38, 116, 419, 727, 030 with the
lefthand declination; and the eight straight lines 024, 327, 630, 125, 425, 7
28, 218, 521 with the righthand abrupt declination, correspond to three types
of spirals on the surface of the cylinder in Fig. 5.17c.
We use the following method of numbering the lattice nodes in Fig. 5.17d,
introducing the following system of coordinates. We use the direct line OO ′ as
the abscissa axis and the vertical trace that passes through the point O as the
ordinate axis. Taking the ordinate of the point 1 as the length unit, the number
that is ascribed to some point of the lattice is then equal to its ordinate. The
lattice that is numbered by the indicated method has some characteristic prop
erties. Any pair of the points gives a certain direction in the lattice system and,
finally, the set of the three parallel directions of the phyllotaxis lattice. We can
see that the lattice in Fig. 5.17d consists of triangles. The vertices of the trian
gles are indicated by the letters a, b and c. It is clear that the lattice in Fig. 5.17
d consists of the set triangles of the kind {c,b,a}, for example, {0,3,8}, {3,6,11},
{3,8,11}, {6,11,14} and so on. It is important to note that the sides of the triangle
{c,b,a} are equal to the differences between the values a,b,c of the triangle {a,b,c}
and are the adjacent Fibonacci numbers: 3,5,8. For example, for the triangle {0,3,8}
we have the following differences: 30=3; 83=5; 80=8. This means that the sides
of the triangle {0,3,8} are equal to, respectively, 3, 5, 8. For the triangle {3,6,11}
we have: 63=3; 116=5; 113=8. This means that its sides are equal to 3, 5, 8,
respectively. Here each side of the triangle defines one of three declinations of
the straight lines that build up the lattice in Fig. 5.17d. In particular, the side of
length 3 defines the righthand small declination, the side of length 5 defines the
lefthand declination, and the side of length 8 defines the righthand abrupt dec
lination. Thus, Fibonacci numbers 3, 5, 8 determine the structure of the phyllo
taxis lattice in Fig. 5.17d.
The second property of the lattice in Fig. 5.17d is the following. The line seg
ment OO′ can be considered as a diagonal of the parallelogram constructed on the
basis of the straight lines corresponding to the lefthand declination and the right
hand small declination. Thus, the given parallelogram allows one to evaluate the
symmetry of the lattice without having to use digital numbering. This parallelo
gram is called a Coordinate Parallelogram. Note that coordinate parallelograms of
different sizes correspond to lattices with different symmetries.
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
309
5.12.2. The Dynamic Symmetry of Phyllotaxis Objects
Here we begin an analysis of the phenomenon of dynamic symmetry. The idea
of analysis involves the comparison of a series of phyllotaxis lattices (the unrolling
of the cylindrical lattice) with different symmetries (Fig. 5.18).
In Fig. 5.18 a variance of Fibonacci phyllotaxis is illustrated, where we ob
serve the following modification of the dynamic symmetry
of the phyllotaxis object during its growth:
1:2:1→ 2:3:1→ 2:5:3 → 3:8:3 → 5:13:8
Note that the lattices, represented in Fig. 5.18, are con
sidered as the sequential stages (5 stages) of the transforma
tion of one and the same phyllotaxis object. There is a ques
tion: how are the transformations of the lattices being car
ried out, that is, which geometric movement can be used to
provide the sequential passing of all illustrated stages of the
phyllotaxis lattice?
5.12.3. The Key Idea of Bodnar’s Geometry
Here we will not go deeply into Bodnar’s original
reasoning that resulted in a new geometrical theory of
phyllotaxis, but direct readers to the remarkable Bod
nar books [37, 52] for a more detailed acquain
tance with his original geometry. Rather we
focus our attention solely on two key ideas
that underlie his geometry.
We first begin with an analysis of the phe
nomenon of dynamic symmetry. The idea of
analysis involves a comparison
of the series of the phyllotaxis
lattices of different symmetries
(Fig. 5.18). We start with the
comparison of stages I and II. At
these stages the lattice can be
transformed by the compres
sion of the plane along the 03
direction up to the position
Figure 5.18. An analysis of dynamic symmetry of a where the line segment 03 ar
rives at the edge of the lattice.
phyllotaxis object
Alexey Stakhov
THE MATHEMATICS OF HARMONY
310
Simultaneously, the expansion of the plane should occur in the 12 direction
perpendicular to the compression direction. At the passing on from stage II to
stage III, the compression should be made along the O5 direction and the ex
pansion along the perpendicular 23 direction. The next passage is accompanied
by similar deformations of the plane in the O8 direction (compression) and in
the perpendicular 35 direction (expansion).
But we know from the prior consideration that the compression of a plane to
any straight line with the coefficient k and the simultaneous expansion of a plane
in the perpendicular direction with the same coefficient k are nothing but Hy
perbolic Rotation. A scheme of hyperbolic transformation of the lattice fragment
is presented in Fig. 5.19. The scheme corresponds to stage II of Fig. 5.18. Note
that the hyperbola of the first quadrant has the equation xy=1 and the hyperbo
la of the fourth quadrant has the equation xy=1.
The transformation of the phyllotaxis lattice in the process of its growth is
carried out by means of the
hyperbolic rotation, the main
yy
xy = 1
geometric transformation of Y
X
hyperbolic geometry, follow
ing from a consideration of the
first key idea of Bodnar’s ge
ometry.
This transformation is
accompanied by a modifica
tion of dynamic symmetry,
x
which can be simulated by
the sequential passage from
the object with smaller sym
metry order to the object
xy = 1
with larger symmetry order.
However, this idea does Figure 5.19. The general scheme of phyllotaxis lattice
not give the answer to the transformation in a system of equatorial hyperboles
question: why are the phyllo
taxis lattices in Fig. 5.18 based on Fibonacci numbers?
5.12.4. The “Golden” Hyperbolic Functions
For a more detailed study of the metric properties of the lattice in Fig. 5.19
we can examine its fragment represented in Fig. 5.20. Here the disposition of the
points is similar to Fig. 5.19.
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
311
Let us pay attention to the basic peculiarities of the disposition of points in
Fig. 5.20: (1) the points M1 and M2 are symmetrical relative to the bisector of the
right angle YOX; (2) the geometric figures OM1M2N1, OM2N2N1, and OM2M3N2
are parallelograms; and (3) the point А is the vertex of the hyperbola xy=1, that
is, xA=1, yA=1, therefore OA = 2.
Y
Let us evaluate the abscissa of
point M2 denoted by x M2 = x . Tak
ing into consideration the symmetry
x
of points M1 and M 2, we can write:
x
1
x M1 = x −1 . It follows from the symme
try condition of these points that the
M
√2
line segment M1M2 is tilted to the coor
√2
M
dinate axis under the angle of 45°. The
M
X
O
line segment M1M2 is parallel to the line
1
segment ON1; this means that the line
N
N
segment ON1 is tilted to the coordinate
axis under the angle of 45°. Therefore, the
point N1 is the top of the lower branch
of the hyperbola; here x N1 = 1,
Figure 5.20. Analysis of the metric
yN1 = 1, ON 1 = OA = 2 . It is clear
properties of the phyllotaxis lattice
that ON 1 = M1M 2 = 2 . It is obvious
that the difference between the abscissas of the points M1 and M2 is equal to 1.
These considerations result in the following equation for the calculation of
the abscissa of the point M2, that is, x M2 = x :
1
1
2
3
2
1
x − x −1 = 1 or x 2 − x − 1 = 0.
(5.155)
This means that the abscissa x M2 = x is the positive root of the famous “gold
en” algebraic equation (5.155):
1+ 5
.
(5.156)
2
Thus, study of the metric properties of the phyllotaxis lattice in Fig. 5.20 un
expectedly leads us to the golden mean. And this fact is the Second Key Outcome
of Bodnar’s Geometry. This result was used by Bodnar for a detailed study of the
phenomenon phyllotaxis. By developing this idea, Bodnar concluded that for a
mathematical simulation of the phyllotaxis phenomenon we need to use a special
class of the hyperbolic functions, the “Golden” Hyperbolic Functions:
The “golden” hyperbolic sine
x M2 = τ =
Gshn =
τn − τ− n
.
2
(5.157)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
312
The “golden” hyperbolic cosine
τn + τ− n
.
(5.158)
2
Furthermore, Bodnar found a fundamental connection of the “golden” hy
perbolic functions with Fibonacci numbers:
2
F2 k −1 =
Gch ( 2k − 1)
(5.159)
5
Gchn =
2
Gsh ( 2k ) .
(5.160)
5
Using the correlations (5.159) and (5.160), Bodnar gave a very simple expla
nation for the “puzzle of phyllotaxis”: why do Fibonacci numbers occur with such
persistent constancy on the surface of phyllotaxis objects? The main reason is the
fact that the geometry of “Living Nature,” in particular, the geometry of phyllo
taxis, is a nonEuclidean geometry; however, this geometry differs substantially
from Lobachevsky’s geometry and Minkowski’s fourdimensional world based on
the classical hyperbolic functions. This difference consists in the fact that the main
correlations of this geometry are described with the help of the “golden” hyperbol
ic functions (5.157) and (5.158) that are connected with Fibonacci numbers by
the simple correlations (5.159) and (5.160).
It is important to emphasize that Bodnar’s model of the dynamic symme
try of the phyllotaxis object illustrated in Fig. 5.20 is brilliantly confirmed by
reallife phyllotaxis pictures of botanic objects.
F2 k =
5.12.15. The Connection of Bodnar’s “Golden” Hyperbolic Functions
with the Hyperbolic Fibonacci Functions
Comparing the expressions for symmetric hyperbolic Fibonacci and Lucas
sines and cosines given by formulas (5.57) (5.60) with expressions for Bodnar’s
“golden” hyperbolic functions given by formulas (5.155) and (5.156), we discov
er the following simple correlations between the indicated groups of the formu
las:
Gsh ( x ) =
5
sFs ( x )
2
(5.161)
Gch ( x ) =
5
cFs ( x )
2
(5.162)
Gsh ( x ) = 2sLs ( x )
(5.163)
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
313
Gch ( x ) = 2cLs ( x ) .
(5.164)
The analysis of these correlations allows us to conclude that the “golden” hy
perbolic sine and cosine introduced by Oleg Bodnar [37], and the symmetric
hyperbolic Fibonacci and Lucas sines and cosines introduced by Stakhov and
Rozin [106], coincide within constant coefficients. The question of the use of
the “golden” hyperbolic functions or the hyperbolic Fibonacci and Lucas func
tions for the simulation of phyllotaxis objects has no particular significance
because the final result is the same: it always results in an unexpected appear
ance of Fibonacci or Lucas numbers on the surfaces of phyllotaxis objects.
5.13. Conclusion
1. The discovery of Lobachevsky’s geometry became an epochmaking
event in the development not only mathematics, but also of science in general.
Academician Kolmogorov appreciated the role of this discovery in the devel
opment of mathematics in the following words [1]: “... It is difficult to overrate the
importance of the reorganization of the entire warehouse of mathematical think
ing, which happened in the 19th century. In this connection, Lobachevsky’s
geometry was the most significant mathematical discovery at the start of the
19th century. Based upon this geometric insight, the belief in the absolute stabil
ity of mathematical axioms was overthrown. This allowed for the creation of essen
tially new and original abstract mathematical theories and, at last, to demonstrate
that similar abstract theories can result hereafter in wider and more concrete ap
plications.” After Lobachevsky’s discovery, the “hyperbolic ideas” started to pen
etrate widely into various spheres of science. After the promulgation of the special
theory of relativity by Einstein in 1905 and its “hyperbolic interpretation,” given
by Minkowski in 1908, the “hyperbolic ideas” became universally recognized. Thus,
a comprehension of the “hyperbolic character” of the processes in the physical world
surrounding us became the major result in the development of science during the
19th and 20th centuries.
The mathematical correlations of Lobachevsky’s geometry are or course based
upon the classical hyperbolic functions. Why did Lobachevsky use these func
tions, introduced by Vincenzo Riccati in the late 18th century, in his geometry?
Apparently, Lobachevsky understood that these functions provide the best way
to model the “hyperbolic character” of his geometry; however, he used them be
Alexey Stakhov
THE MATHEMATICS OF HARMONY
314
cause other hyperbolic functions at that moment simply did not exist. It is nec
essary to note that Lobachevsky’s geometry, based on classical hyperbolic func
tions, is historically the first “hyperbolic model” of physical space. Lobachev
sky’s geometry and Minkowski’s geometry had put forward the hyperbolic func
tions as the basic plan for modern science.
2. At the end of the 20th century, the Ukrainian architect Oleg Bodnar [37]
and the Ukrainian mathematicians Alexey Stakhov and Ivan Tkachenko [98]
broke the monopoly of classical hyperbolic functions in modern science. They
introduced a new class of hyperbolic functions based on the golden mean. Later
Alexey Stakhov and Boris Rozin developed the symmetrical hyperbolic Fibonacci
and Lucas functions [108]. However, Bodnar, Stakhov, Tkachenko and Rozin
each used their own unique methods to arrive at a new class of hyperbolic func
tions. Oleg Bodnar found the “golden” hyperbolic functions thanks to his scien
tific intuition, which resulted in the “golden” hyperbolic functions as the “key”
idea in the study of the phyllotaxis phenomenon. These functions were used by
him for the creation of the original geometric theory of phyllotaxis (Bodnar’s
Geometry). The approach of Alexey Stakhov, Ivan Tkachenko and Boris Rozin
was based on an analogy between the Binet formulas and hyperbolic functions.
This approach resulted in the discovery of a new class of hyperbolic functions,
Hyperbolic Fibonacci and Lucas Functions.
3. The hyperbolic Fibonacci and Lucas functions are a generalization of the
Fibonacci and Lucas numbers “extended” to the continuous domain. There is a
direct analogy between the Fibonacci and Lucas number theory and the theo
ry of hyperbolic Fibonacci and Lucas functions because the “extended” Fibonac
ci and Lucas numbers coincide with the hyperbolic Fibonacci and Lucas func
tions at discrete values of the variable x (x=0, ±1, ±2, ±3, ...). Besides, every
identity for the Fibonacci and Lucas numbers has its continuous analog in the
form of a corresponding identity for the hyperbolic Fibonacci and Lucas func
tions, and conversely. This outcome is of special significance for the Fibonacci
number theory [13, 16, 28, 38] because this theory is as if it were transformed
into the theory of hyperbolic Fibonacci and Lucas functions [98, 108, 116, 119].
Thanks to this approach, the Fibonacci and Lucas numbers became one of the
most important numerical sequences of hyperbolic geometry.
4. Bodnar’s geometry [37, 52] demonstrates that the “golden” hyperbolic
world exists objectively and independently of our consciousness and persis
tently appears in Living Nature, in particular, in pine cones, pineapples, cacti,
heads of sunflowers and baskets of various flowers in the form of Fibonacci and
Lucas spirals on the surface of these biological objects (the phyllotaxis law).
Furthermore Bodnar’s geometry is a demonstration of the “physical practica
Chapter 5
Hyperbolic Fibonacci and Lucas Functions
315
bility” of the hyperbolic Fibonacci and Lucas functions, which underlie all of
Living Nature. However, the promulgation of the new geometrical theory of
phyllotaxis, made by the Ukrainian architect Oleg Bodnar [37], had demon
strated that in addition to “Lobachevsky’s geometry.” Nature also uses other
variants of the socalled “hyperbolic models of Nature” [98, 108]. The use of
hyperbolic Fibonacci and Lucas functions allowed for the solution to the “rid
dle of phyllotaxis,” that is, an explanation of, how Fibonacci and Lucas spirals
appear on the surface of phyllotaxis objects.
5. However, perhaps the final step in the development of the “hyperbolic
models” of Nature was made by Alexey Stakhov in his article [118]. The hyper
bolic Fibonacci and Lucas mfunctions are a wide generalization of the symmet
ric hyperbolic Fibonacci and Lucas functions introduced by Stakhov and Rozin
in an earlier article [106]. They are based on the Gazale formulas and extend ad
infinitum a number of new hyperbolic models of Nature. It is difficult to imagine
that the set of new hyperbolic functions is infinite! The hyperbolic Fibonacci
and Lucas mfunctions do complete a general theory of hyperbolic functions,
started by Johann Heinrich Lambert and Vincenzo Riccati, and opens new per
spectives for the development of new “hyperbolic ideas” in modern science.
The development of a general theory of hyperbolic functions [118], based
on the Gazale formulas [45], gives us the opportunity to put forward the fol
lowing unusual hypothesis. Apparently, we can assume that theoretically there
are an infinite number of “hyperbolic models of Nature.” One of the possible
types of the hyperbolic functions given by the general formulas (5.105) (5.108)
underlies these models. Each type of hyperbolic function meets some positive
real number m, called the order of the function. This number generates new
mathematical constants, the generalized golden mproportions, given by the
formula (4.266). Each type of hyperbolic function, in turn, generates new classes
of the recursive numerical sequences, the generalized Fibonacci and Lucas m
numbers given by the recursive relations (4.251) and (4.302).
Nature intelligently uses one or another type of hyperbolic function for
introducing objects into this or that “hyperbolic world.” It is necessary to note
that “Lobachevsky’s geometry” is one of the possible variants of the realization
of the “hyperbolic world,” which probably is preferable for the objects of “min
eral Nature.” Apparently in the case of “Living Nature,” “Bodnar’s geometry”
based on the hyperbolic Fibonacci and Lucas functions, is preferable. This fact
is confirmed by the “law of phyllotaxis” met in many botanical objects. But
“Lobachevsky’s geometry” and “Bodnar’s geometry” are not unique variants
of the realization of the “hyperbolic worlds.” It is possible to expect that new
types of hyperbolic functions, described in [118], will result in the discovery of
Alexey Stakhov
THE MATHEMATICS OF HARMONY
316
new “hyperbolic worlds” which can be embodied in natural structures at dif
ferent levels of organization in the Universe. Importantly, the occurrence of
characteristic recursive numerical sequences given by the general recursive for
mulas (4.251) and (4.302) are “external attributes”, embodied in Nature’s “hy
perbolic worlds,” found in the search for corresponding types of hyperbolic
functions (5.105) (5.108).
Chapter 6
Fibonacci and Golden Matrices
317
Chapter 6
Fibonacci and Golden Matrices
6.1. Introduction to Matrix Theory
6.1.2. A History of Matrices
The history of matrices goes back to ancient times. But the term “matrix”
was not applied to the subject until 1850 by James Joseph Sylvester. “Matrix” is
the Latin word for womb, and it retains that sense in English. It can also mean,
more generally, any place in which something is formed or produced.
The origin of mathematical matrices arose with the study of systems of
simultaneous linear equations. An important Chinese text from between
300 BC and 200 AD, Nine Chapters of the Mathematical Art (Chiu Chang
Suan Shu), is the first known example of the use of matrix methods to solve
simultaneous equations. In the treatise’s seventh chapter, Too Much and
Not Enough, the concept of a determinant first appears, nearly two millen
nia before its supposed invention by the Japanese mathematician Seki Kowa
in 1683, or alternatively, his German contemporary Gottfried Leibnitz (who
is also credited with the invention of differential calculus, independently
though simultaneously with Isaac Newton). More uses of matrixlike ar
rangements of numbers appear in chapter eight, Methods of Rectangular
Arrays, in which a method is given for solving simultaneous equations. This
method is mathematically identical to the modern matrix method of solu
tion outlined by Carl Friedrich Gauss (17771855), also known as Gauss
ian elimination.
James Joseph Sylvester (1814 – 1897) was an English mathematician.
He made fundamental contributions to matrix theory, invariant theory, num
ber theory, partition theory and combinatorics. He played a leading role in
American mathematics in the latter half of the 19th century as a professor at
Johns Hopkins University and as founder of the American Journal of Mathe
matics. At his death, he was a professor at Oxford.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
318
Sylvester began his study of mathematics at St.
John’s College, Cambridge in 1831. In 1838 he be
came professor of natural philosophy at University
College London. In 1841 he moved to the United
States to become a professor at the University of
Virginia, but soon returned to England. On his re
turn to England he studied law, alongside British
lawyer/mathematician Arthur Cayley, with whom
he made significant contributions to matrix theory.
James Joseph Sylvester
Poetry was one of Sylvester’s lifelong passions; he
(1814 1897)
read and translated works out of the original French,
German, Italian, Latin and Greek, and many of his mathematical papers
contain illustrative quotes from classical poetry. In 1870, following his ear
ly retirement, Sylvester published a book titled The Laws of Verse in which
he attempted to codify a set of poetry laws. In 1877 he again crossed the
Atlantic to become the inaugural professor of mathematics at the new Johns
Hopkins University in Baltimore, Maryland. In 1878 he founded the Amer
ican Journal of Mathematics. And in 1883, Sylvester returned to England
to become Savilian Professor of Geometry at Oxford University.
Since their first appearance in ancient China, matrices have remained
important mathematical tools. Today, they are used not only for solving
systems of simultaneous linear equations, but also for describing quantum
atomic structure, designing computer game graphics, analyzing relation
ships, and in numerous other capacities.
The elevation of the matrix from mere tool to important mathematical
theory owes a lot to the work of female mathematician Olga Taussky Todd
(19061995), who began by using matrices to analyze vibrations on airplanes
and became the torchbearer for matrix theory.
6.1.2. Definition of a Matrix
In mathematics, a Matrix (plural Matrices) is a rectangular table of
numbers or, more generally, a table consisting of abstract quantities that
can be added and multiplied. Matrices are used to describe linear equa
tions, keep track of the coefficients of linear transformations, and to
record data that depend on two parameters. Matrices can be added,
multiplied, and decomposed in various ways, making them a key concept
in linear algebra and matrix theory [157].
Chapter 6
Fibonacci and Golden Matrices
319
Let us consider, for example, a matrix A
 a11 a12 " a1n 
 a a " a2n 
.
A =  21 22
#
# # # 
a a " a 
mn 
 m1 m 2
The horizontal lines in a matrix are called Rows and the vertical lines are
called Columns. A matrix with m rows and n columns is called an mbyn ma
trix (written m×n) and m and n are called its Dimensions. The dimensions of a
matrix are always given with the number of rows first, then the number of
columns. The entry of a matrix A that lies in the ith row and the jth column
is called the (i, j)th entry of A. This is written as Ai,j or A[i, j]. As is indicated,
the row is always noted first, then the column.
A matrix, where one of the dimensions is equal to 1, is often called a Vector.
The (1×n)matrix (one row and n columns) is called a Row Vector, and the
(m×1)matrix (m rows and one column) is called a Column Vector.
6.1.3. Matrix Addition and Scalar Multiplication
Let A and B be two matrices with the same size, i.e. the same number of
rows and of columns. The sum of A and B, written A+B is the matrix obtained
by adding corresponding elements from A and B:
 a11 a12 ... a1n   b11 b12 ... b1n   a11 + b11 a12 + b12 ... a1n + b1n 
 a21 a22 ... a2 n   b21 b22 ... b2 n   a21 + b21 a22 + b22 ... a2n + b2n 
 ... ... ... ...  +  ... ... ... ...  =  ...
.
...
...
...
 a a ... a   b b ... b   a + b a + b ... a + b 
 m1 m 2
 m1 m 2
 m1 m1 m 2 m 2
mn
mn 
mn 
mn 
The product of a scalar k and a matrix A, written kA or Ak is the matrix
obtained by multiplying each element of A by k:
 a11 a12 ... a1n   ka11 ka12 ... ka1n 
 a21 a22 ... a2n   ka21 ka22 ... ka2n 
k
=
.
... ... ... ...   ...
... ... ... 
 a a ... a   ka ka ... ka 
mn 
mn  
m1
m2
 m1 m 2
We also define:
A=(1)A and AB=A+(B)
Note that the matrix (A) is negative to the matrix A.
Let us consider the following theorem for matrix addition and scalar
multiplication.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
320
Theorem 6.1. Let A, B and C be matrices with the same size and let k and
k′ be scalars. Then we have:
( A + B ) + C = A + ( B + C ) , i.e., matrix addition is associative;
( A + B ) = (B + A), i.e., matrix addition is commutative;
A + 0 = 0 + A; A + ( − A ) = 0; k ( A + B ) = kA + kB;
( k + k′) A = kA + k′ A; ( k × k′) A = k ( k′ A ) ; 1× A = A.
6.1.4. Matrix Multiplication
Suppose A and B are two matrices such that the number of columns of A is
equal to the number of rows of B, say A is an (m×p)matrix and B is a (p×n)
matrix. Then the product of A and B, written AB is the (m×n)matrix whose
(i, j)entry is obtained by multiplying the elements of the ith row of A by the
corresponding elements of the jth column of B and then adding:
 a11 ... a1 p   b11 ... b1 j ... b1n   c11 ... c1n 
 ... ... ...   ... ... ... ... ...   ... ... ... 
 ai1 ... aip   ... ... ... ... ...  =  ... cij ...  ,
 ... ... ...   ... ... ... ... ...   ... ... ... 
 am1 ... amp   bp1 ... bpj ... bpn   cm1 ... cmn 

 


where
p
cij = ai1b1 j + ai 2b2 j + ... + aip bpj =
∑a b .
ik kj
k =1
Matrix multiplication satisfies the following properties:
Theorem 6.2.
( AB)C = A ( BC ) ; A ( B + C ) = AB + AC ;
k ( AB ) = ( kA) B = A ( kB ) ,
where k is a scalar.
6.1.5. Square Matrices
6.1.5.1. Definition of a Square Matrix
A matrix with the same number of rows and columns is called a Square
Matrix. A square matrix with n rows and n columns is said to be of order n, and
is called an nsquare matrix. The Main Diagonal, or simply Diagonal, of a
square matrix A=(aij) consists of the numbers:
(a11, a22,…, ann) .
Chapter 6
Fibonacci and Golden Matrices
321
The nsquare matrix with 1’s along the main diagonal and 0’s elsewhere, e.g.
1 0 0 0
I =  0 1 0 0 
0010
0 0 0 1


is called the Unit or Identity Matrix and will be denoted by I. The unit matrix I
plays the same role in matrix multiplication as the number 1 does in the usual
multiplication of numbers.
Specifically,
AI=IA=A
for any square matrix A.
Square matrices can be raised to various powers. We can define powers of
the square matrix A as follows:
A2=AA, A3= A2A, …, and A0=I.
6.1.5.2. Invertible Matrices
A square matrix A is said to be Invertible if there exists a matrix B with the
property that
AB=BA=I.
Such a matrix B is unique; it is called the Inverse of A and is denoted by A1.
6.1.5.3. Determinants
To each nsquare matrix A=(aij) we assign a specific number called the
Determinant of A, denoted by Det(A) or A.
The determinants of order one, two and three are defined as follows:
Det ( a11 ) = a11 = a11
a a
Det  11 12  = a11a22 − a12a21
a
 21 a22 
a11 a12 a13
a21 a22 a23 = a11a22 a33 + a12 a23 a31 + a13 a21a32 − a13 a22 a31 − a12 a21a33 − a11a23 a32
a31 a32 a33
An important determinant property is given by the following theorems.
Theorem 6.3. For any two nsquare matrices A and B we have
Det(A×B)=DetA×DetB
Theorem 6.4
Det(An)=(DetA)n.
(6.1)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
322
6.2. Fibonacci QMatrix
6.2.1. A Definition of the QMatrix
In recent decades the theory of Fibonacci numbers [13, 16, 28] has been sup
plemented by the theory of the socalled Fibonacci Qmatrix [9]. The latter is
the (2×2)matrix of the following form:
( )
Q= 11 .
10
(6.2)
Note that the determinant of the Qmatrix is equal to 1:
DetQ=1.
(6.3)
The article [158] devoted to the memory of Verner E. Hoggatt, founder of
the Fibonacci Association, contained the history and an extensive bibliogra
phy of the Qmatrix and emphasized Hoggatt’s contribution in its develop
ment. Although the name of the Qmatrix was introduced before Verner E.
Hoggatt, it was from Hoggatt’s articles that the idea of the Qmatrix “caught
on like wildfire among Fibonacci enthusiasts. Numerous papers have appeared
in ‘The Fibonacci Quarterly’ authored by Hoggatt and/or his students and
other collaborators where the Qmatrix method became a central tool in the
analysis of Fibonacci properties” [158].
We will consider here a theory of the Qmatrix developed in Hoggatt’s
book [16].
6.2.2. Properties of the QMatrix
The following theorem connects the Qmatrix to Fibonacci numbers.
Theorem 6.5. For a given integer n the nth power of the Qmatrix is given by
F
F
Q n =  n +1 n  ,
F
F
 n n −1 
where Fn1, Fn, Fn+1, are Fibonacci numbers.
Proof. We use mathematical induction. Clearly, for n=1,
F F
Q 1 =  2 1  =  1 1  .
 F1 F0   1 0 
F 
F
k
Suppose that Q =  Fk +1 F k  , then
 k k −1 
(6.4)
Chapter 6
Fibonacci and Golden Matrices
323
F  1 1   F + F F   F F 
F
Q k +1 = Q k ×Q =  k +1 k  ×   =  k +1 k k +1  =  k + 2 k +1  .
 Fk Fk −1   1 0   Fk + Fk −1 Fk   Fk +1 Fk 
The theorem is proved.
The next theorem gives a formula for the determinant of the matrix (6.5).
Theorem 6.6. For a given integer n we have:
(6.5)
Det(Qn)=(1)n.
Proof. Using (6.1) and (6.5), we can write:
Det(Qn)=( DetQ)n=(−1)n.
The theorem is proved.
The following remarkable property for Fibonacci numbers follows from
Theorem 6.5:
n
(6.6)
DetQ n = Fn −1Fn +1 − Fn2 = ( −1) .
Recall that the identity (6.6) is one of the most important identities for
Fibonacci numbers. This one is called the Cassini formula in honor of the
famous French astronomer Giovanni Domenico Cassini (16251712), who
discovered this formula for the first time.
Now, represent the matrix (6.4) in the following recursive form:
 F + Fn −1 Fn −1 + Fn −2   Fn Fn −1   Fn −1 Fn − 2 
(6.7)
Qn =  n
=
+

 Fn −1 + Fn −2 Fn −2 + Fn −3   Fn −1 Fn −2   Fn − 2 Fn − 3 
or
(6.8)
Q n = Q n −1 + Q n −2 .
We can represent the recursive relation (6.8) in the following form:
n −2
(6.9)
Q = Q n − Q n −1.
n
n
The explicit forms of the matrices Q and Q ( n = 0, ±1, ±2, ±3,... ) obtained
by means of the use of the recursive relations (6.8) and (6.9) are
given in Table 6.1.
Table 6.1. Qmatrices
n
Q
n
Q −n
0
1
1 1
1 0
0 1
1 −1
1
2
3
4
5
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
1
0
1
0
0
1
0
1
2
1
1
−1
1
1
−1
2
3
2
−1
2
2
1
2
−3
5
3
2
−3
3
2
−3
5
8
5
−3
5
5
3
5
−8
6.2.3. Binet Formulas for the QMatrix
We can find the analytical expressions for the following sum and difference:
Qn+Qn and Qn−Qn.
(6.10)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
324
We can start from the following examples for the even (n=2k) and odd
(n=2k+1) powers. For the case n=5 we have the following expression for the
matrix sum (6.10):
( )(
)(
) ( )
Q 5 + Q −5 = 8 5 + −3 5 = 5 10 = 5 1 2 = F5T ,
5 3
5 −8
10 −5
2 −1
(6.11)
where F5 is a Fibonacci number and
( )
T= 1 2 .
2 −1
(6.12)
We have the following expression for the matrix difference (6.10):
( )(
)( ) ( )
Q 5 − Q −5 = 8 5 − −3 5 = 11 0 = 11 1 0 = L5 I ,
5 3
5 −8
0 11
0 1
(6.13)
where L5 is a Lucas number, I is the identity matrix.
For the case n=6 we have:
( )( )( ) ( )
Q − Q = (13 8 ) − ( 5 −8 ) = ( 8 16 ) = 8 ( 1 2 ) = F T ,
−8 13
8 5
16 −8
2 −1
Q 6 + Q −6 = 13 8 + 5 −8 = 18 0 = 18 1 0 = L6 I
−8 13
8 5
0 18
0 1
6
−6
6
(6.14)
(6.15)
where the matrix T is given by (6.12).
The expressions (6.11), (6.13), (6.14), (6.15) are the partial cases of the
following general formulas valid for an arbitrary n (n=0, ±1, ±2, ±3,…):
Q2k+1 Q(2k+1)=L2k+1I
(6.16)
Q2k Q2k=L2kI
(6.17)
Q2k+1+ Q(2k+1)=F2k+1T
(6.18)
Q2k+ Q2k=F2k+1T
(6.19)
Note that the formulas (6.16) (6.19) are the matrix equivalents of the
Binet formulas (2.67) and (2.68). The formulas (6.16) and (6.17) are the ma
trix equivalent of the Binet formula (2.67) for Lucas numbers and the formu
las (6.18) and (6.19) are the matrix equivalents of the Binet formula (2.68)
for Fibonacci numbers. It is clear that the Qmatrix (6.2) in these formulas
plays the role of the golden ratio τ in the formulas (2.67) and (2.68). For all
that, the special matrix (6.12) plays the role of the irrational number 5 in
the Binet formula (2.68) for Fibonacci numbers.
This analogy between the golden ratio τ and Qmatrix (6.2) shows
that there is an isomorphism between the golden mean theory and the Q
matrix theory. We can prove the following theorems confirming this isomor
Chapter 6
Fibonacci and Golden Matrices
325
phism. First of all, we can write the following trivial property for the golden
ratio powers:
τnτm =τmτn =τn+m .
Note that we do not need to prove for the number τ the following obvious
equality: τnτm =τmτn, however, for the matrices we need to prove the similar
equality.
We can prove the following theorem for the Qmatrices.
Theorem 6.7.
QnQm =QmQn =Qn+m .
Proof. At first, we prove the identity
(6.20)
QnQm = Qn+m .
In order to prove this identity we can write the product of the matrices QnQm
as follows:
Fn   Fm +1 Fm 
 Fn +1
Q nQ m = 
×

 F

Fm Fm −1 
F
n −1  
 n
F F +F F
=  n +1 m +1 n m
 Fn Fm +1 + Fn −1Fm
Fn +1Fm + Fn Fm −1 
.
Fn Fm −1 + Fn −1Fm −1 
(6.21)
For the proof we use the formula for the generalized Fibonacci numbers Gn
described in Section 2.5:
Gn+m=Gn+1Fm+GnFm1 .
If Gi=Fm , the formula (6.22) takes the following form:
(6.22)
(6.23)
Fn+m=Fn+1Fm+FnFm1.
Using (6.23), we can represent the elements of the matrix (6.21) as follows:
Fn+1Fm+1+FnFm= Fn+m+1
(6.24)
Fn+1Fm+FnFm1= Fn+m
(6.25)
FnFm+1+Fn1Fm= Fn+m
(6.26)
FnFm1+Fn1Fm1= Fn+m1.
(6.27)
Taking into consideration the formulas (6.24) (6.27), we can represent the
matrix (6.21) as follows:
F
F

Q nQ m =  n + m +1 n + m  = Q n + m .
 Fn + m Fn + m −1 
By analogy we can prove the validity of the following identity:
QmQn = Qn+m .
Alexey Stakhov
THE MATHEMATICS OF HARMONY
326
It follows from the identity (6.20) that the matrices Qn and Qm possess the
property of multiplication commutativity.
Thus, the above property of the Qmatrix, given by (6.20), gives us the right
1 1
to affirm that the Fibonacci matrix Q = 1 0 is a special matrix, which plays in
the theory of twobytwo matrices a particular role similar to the role of the
golden mean in the theory of real numbers.
( )
6.3. Generalized Fibonacci QpMatrices
6.3.1. A Definition of the QpMatrices
Note that the Qmatrix is a generating matrix for the classical Fibonacci
numbers given by the recursive relation (2.3). In the article [103] Alexey
Stakhov generalized the concept of the Qmatrix [16, 158] and introduced
the generalized Fibonacci Qpmatrices. For a given p (p=1, 2, 3, …) the gener
alized Fibonacci Qpmatrix has the following form:
1 1 0 0 " 0 0
0 0 1 0 " 0 0
0 0 0 1 " 0 0
Qp =  # # # # # # # 
0 0 0 0 " 1 0
(6.28)
0 0 0 0 " 0 1


1 0 0 0 " 0 0
The Qpmatrix (6.28) is a square (p+1)×(p+1)matrix. It consists of the
identity (p×p)matrix bordered by the last row consisting of 0’s and the lead
ing 1, and the first column consisting of 0’s embraced by a pair of 1’s. We can
list the Qp matrices for the case p=1, 2, 3, … as follows:
1 1 0 0 0
1 1 0 0
1 1 0
0 0 1 0 0

11
0 0 1 0
Q1 =
; Q4 =  0 0 0 1 0  .
= Q ; Q2 =  0 0 1  ; Q 3 = 
10
0 0 0 1
1 0 0
0 0 0 0 1
1 0 0 0


1 0 0 0 0




( )
(6.29)
6.3.2. The Main Theorems for the QpMatrices
Next let us raise the Qpmatrix (6.28) to the nth power and find the
analytical expression for the matrix Q pn . Let us prove the following theorem.
Theorem 6.8. For a given integer p= 1, 2, 3, … we have:
Chapter 6
Fibonacci and Golden Matrices
327
Fp (n) " Fp (n − p + 2) Fp (n − p + 1) 
 Fp (n + 1)
 Fp (n − p + 1) Fp (n − p) " Fp (n − 2 p + 2) Fp (n − 2 p + 1) 
,
Q pn = 
#
#
#
#
#
 F (n − 1) F (n − 2) " F (n − p)
(6.30)
Fp (n − p − 1) 
p
p
 p

Fp (n − 1) " Fp (n − p + 1)
F p (n − p ) 
 F p (n)
where Fp(n) is the Fibonacci pnumber, n=0, ±1, ±2, ±3, … .
Before the proof of Theorem 6.8 we analyze the matrix (6.30). Note that all
entries of the matrix are Fibonacci pnumbers given by the recursive relation
(4.18) at the seeds (4.19). The matrix (6.30) consists of (p+1) rows and (p+1)
columns. The first row is the following sequence of the Fibonacci pnumbers:
Fp(n+1), Fp(n),…, Fp(np+2), Fp(np+1),
(6.31)
the second row consists of the Fibonacci pnumbers:
(6.32)
Fp(np+1), Fp(np),…, Fp(n2p+2), Fp(n2p+1),
the pth row consists of the Fibonacci pnumbers:
(6.33)
Fp(n1), Fp(n2),…, Fp(np), Fp(np1),
the (p+1)th row consists of the following Fibonacci pnumbers:
(6.34)
Fp(n), Fp(n1),…, Fp(np+1), Fp(np).
Note that every one of the rows given by (6.31) – (6.34) is the sequence of
the Fibonacci pnumbers consisting of (p+1) sequential Fibonacci pnumbers.
The first row (6.31) begins with the Fibonacci pnumber Fp(n+1) and finishes
with the Fibonacci pnumber Fp(np+1), the second row (6.32) begins with the
Fibonacci pnumber Fp(np+1) and finishes with the Fibonacci pnumber Fp(n
2p+1) and finally, the (p+1)th row begins from the Fibonacci pnumber Fp(n)
and finishes with the Fibonacci pnumber Fp(np). It is important to emphasize
that the second Fibonacci pnumber of the first row (6.31) and the first Fi
bonacci pnumber of the last row (6.34) are equal to Fp(n), the third Fibonacci
pnumber of the second row (6.32), the fourth Fibonacci pnumber of the third
row, ..., and the (p+1)th Fibonacci pnumber of the pth row (6.33) are equal to
Fp(np1). Now we can start a proof of Theorem 6.8.
Proof. For a given p we can use the induction method.
(a) The basis of the induction. At first we prove the basis of the induction,
that is, we prove that the statement (6.30) is valid for the case n=1. It is clear
that for the case n=1 the matrix (6.30) takes the following form:
Fp (1) " Fp (− p + 3) Fp (− p + 2) 
 Fp (2)
 Fp (− p + 2) Fp (− p + 1) " Fp (−2 p + 3) Fp (−2 p + 2) 

.
#
#
#
#
#
 F (0)

(
−
1
)
"
(
−
+
1
)
(
−
)
F
F
p
F
p
p
p
p
p


Fp (0) " Fp (− p + 2) Fp (− p + 1) 
 Fp (1)
(6.35)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
328
Taking into consideration the values of the initial terms of the Fibonacci p
numbers given by (4.19) and the mathematical properties of the extended Fi
bonacci pnumbers given by (4.29), (4.31) and (4.32) we can write:
Fp(2)=Fp(1)=Fp(p)=1.
Note that all the remaining entries of the matrix (6.34) are equal to 0. This
means that the matrix (6.35) coincides with the Qpmatrix given by (6.28). The
basis of the induction is proved.
(b) Inductive hypothesis. Suppose that the statement (6.30) is valid for
arbitrary n, and then prove it for n=1. For this purpose we shall consider the
product of the matrices
 1 1 0 0 " 0 0
Fp ( n) " Fp ( n − p + 2) Fp ( n − p + 1) 
 Fp (n + 1)
 0 0 1 0 " 0 0
Fp ( n − p + 1) Fp ( n − p) " Fp (n − 2 p + 2) Fp (n − 2 p + 1)

 0 0 0 1 " 0 0
n
Qp ×Qp = 
#
#
#
#
#
 ×  # # # # % # #  . (6.36)
 Fp (n − 1) Fp (n − 2) " Fp (n − p) Fp (n − p − 1)   0 0 0 0 " 1 0
 F (n)

Fp ( n − 1) " Fp ( n − p + 1)
Fp ( n − p )   0 0 0 0 " 0 1
p
 1 0 0 0 " 0 0
Let us now consider the matrix (6.36). If we multiply the first row of the
matrix Q pn by the matrix Qp in the example (6.36) we obtain the entry a11:
a11= Fp(n+1)+ Fp(np+1)= Fp(n+2).
Continuing the process of the matrix multiplication for the example
(6.36), we obtain the following entries of the first row of the matrix Q pn+1 :
Fp(n+2), Fp(n+1),…, Fp(np+3), Fp(np+2).
(6.37)
Next let us compare the first row of the matrix Q pn given by (6.31) with the
first row of the matrix Q pn+1 given by (6.37). We can see that all arguments of
the Fibonacci pnumbers in the sequence (6.37) differ by 1 from the arguments
of the corresponding Fibonacci pnumbers, which form the first row (6.31) of
the matrix (6.30). By analogy we can show that this rule is valid for all entries of
the matrix Q pn+1 , which is formed from the matrix Q pn by means of its multipli
cation by the matrix Qp in accordance with (6.36). This consideration proves the
validity of Theorem 6.8.
It is easy to prove the following identity for the product of the following
Qp matrices:
Q pn +1 = Q pn × Q p = Q p × Q pn .
Using (6.38), we can prove the following theorem.
Theorem 6.9.
Q pn × Q pm = Q pm × Q pn = Q pm + n .
(6.38)
Chapter 6
Fibonacci and Golden Matrices
329
Decomposing each entry of the matrix (6.30) the Fibonacci pnumber in
accordance with the basic recursive relation (4.18), we obtain the following re
sult given by Theorem 6.10.
Theorem 6.10.
Q pn = Q pn −1 + Q pn − p −1.
(6.39)
This recursive relation (6.39) can also be represented in the following form:
Q pn − p −1 = Q pn − Q pn −1 .
(6.40)
n
For example, consider the matrices Q p corresponding to the cases
p=2 and p=3
 F2 (n + 1)
n

Q2 = F2 (n − 1)
F2 (n) F2 (n − 1) 
F2 (n − 2) F2 (n − 3) 

F2 (n − 1) F2 (n − 2) 

 F2 (n)
 F3 (n + 1) F3 (n) F3 (n − 1) F3 (n − 2) 
 F (n − 2) F (n − 3) F (n − 4) F (n − 5) 
3
3
3
Q3n =  3
.
F
n
F
n
F
n
F
(
−
1
)
(
−
2
)
(
−
3
)
(n − 4) 
3
3
3
3


 F3 (n) F3 (n − 1) F3 (n − 2) F3 (n − 3) 
(6.41)
(6.42)
Using the matrices (6.41) and (6.42) and the general matrix (6.30), we
obtain some special matrices, in particular, the matrices of the kind Q pp and
the inverse matrices of the kind Q p−1. For example, for the cases p=2, 3 , 4, 5
the matrices of the kind Q pp and Q p−1 have the following form:
p=2
 F2 (3)
2
Q2 =  F2 (1)
F2 (2) F2 (1)   1 1 1 
F2 (0) F2 (−1)  =  1 0 0 

 

 F2 (2) F2 (1) F2 (0)   1 1 0 
 F2 (0) F2 (−1) F2 (−2)   0 0 1 
Q2−1 =  F2 (−2) F2 (−3) F2 (−4)  =  1 0 −1

 

 F2 (−1) F2 (−2) F2 (−3)   0 1 0 
p=3
 F3 (4) F3 (3) F3 (2) F3 (1)   1 1 1 1 
 F (1) F (0) F (−1) F (−2)   1 0 0 0 
3
3
3
Q33 =  3
=
F
F
F
(
2
)
(
1
)
(
0
)
1 1 0 0
F
3
3
3
3 (−1) 




 F3 (3) F3 (2) F3 (1) F3 (0)   1 1 1 0 
 F3 (0) F3 (−1) F3 (−2) F3 (−3)   0 0 0 1 
 F (−3) F (−4) F (−5) F (−6)   1 0 0 −1
3
3
3
Q3−1 =  3
=
2
(
)
F
F
F
F
−
(
−
3
)
(
−
4
)
0 1 0 0
3
3
3 (−5) 
3




 F3 (−1) F3 (−2) F3 (−3) F3 (−4)   0 0 1 0 
(6.43)
(6.44)
(6.45)
(6.46)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
330
p=4
0 0 0 0 1 
1 1 1 1 1 
1 0 0 0 0 
 1 0 0 0 −1 
Q44 =  1 1 0 0 0  Q4−1 =  0 1 0 0 0 
1 1 1 0 0 
0 0 1 0 0 
1 1 1 1 0 
0 0 0 1 0 




(6.47)
p=5
1 1 1 1 1 1 
0 0 0 0 0 1 
1 0 0 0 0 0 
 1 0 0 0 0 −1
 1 1 0 0 0 0  −1  0 1 0 0 0 0 
5
.
Q5 = 
Q =
1 1 1 0 0 0 5 0 0 1 0 0 0 
(6.48)
1 1 1 1 0 0 
0 0 0 1 0 0 
1 1 1 1 1 0 
0 0 0 0 1 0 




Looking at the matrices (6.43) (6.48), we can see that they have a strong
regular form. This allows us to construct similar matrices for arbitrary p.
Comparing the matrices of the kind Q pp ( p = 1, 2, 3, ... ) we can see that each ma
trix Q pp includes in itself all the preceding matrices of the kind Q pp−−11 , Q pp−−22 , ..., Q11.
For example, the matrix Q pp−−11 is built up from the matrix Q pp by means of strik
ing out the last row and last column of the matrix Q pp .
The last rows and columns of all matrices of the kind Q pp have a strong
regular structure. In particular, each last row has the following form (111…10)
(p+1 entries) and each last column begins with the top 1, but all the remain
ing entries are equal to zero.
Comparing the inverse matrices of the kind Q p−1 ( p = 1, 2, 3, ... ) , we can find
the following regularity. The main diagonal of the matrix Q p−1 consists of zeros.
All entries, forming the diagonal under the main diagonal, are equal to 1. The last
column starts with the entries 1 and −1. All of the remaining entries of the ma
trix Q p−1 are equal to 0.
−1
There is the following connection between the next matrices Q p−1 and Q p−
1.
−1
−1
The matrix Q p−1 can be obtained from the matrix Q p , if we strike out the last
row and the next to the last column of the matrix Q p−1 .
6.4. Determinants of the QpMatrices and their Powers
Now let us calculate the determinants of the Qpmatrices (6.28). It is
clear that DetQ1=1. Calculate the determinant of the Q2matrix
1 1 0
Q2 =  0 0 1  .
1 0 0


(6.49)
Chapter 6
Fibonacci and Golden Matrices
331
Comparing the Q2matrix (6.49) with the Q1matrix
Q1 = 1 1 = Q ,
(6.50)
1 0
we can see that the Q2matrix (6.49) is reduced to the Q1matrix (6.50) if we strike
out from the Q2matrix (6.49) the 2nd row and the 3rd column. Note that the sum of
these numbers 2+3=5. According to the matrix theory [157] we have:
( )
DetQ2=1×(1)5×DetQ1=1×(1)×(1)=1.
Now let us calculate the determinant of the Q3matrix
(6.51)
1 1 0 0
Q3 =  0 0 1 0  .
0 0 0 1
(6.52)
1 0 0 0


Comparing the matrix (6.52) with the matrix (6.49) we can see that the Q3 matrix (6.52) is reduced to the Q2 matrix (6.49) if we strike out the 3rd row and
the 4th column (3+4=7) in the matrix (6.52). This means [157] that
DetQ3=1×(1)7×DetQ2=1×(1)×1=1.
(6.53)
By analogy we can show that
DetQ4=1×(1)9×DetQ3=1×(1)×(1)=1.
(6.54)
Ccontinuing this process, that is, generating the formulas similar to (6.51),
(6.53), and (6.54) we can write the following recursive relation, which con
nects the determinants of the adjacent matrices Qp and Qp1:
DetQp=1×(1)2p+1×DetQp1.
(6.55)
It is easy to show that DetQp is equal to 1 for the even values of p=2k and
is equal to (1) for the odd values of p=2k+1. We can formulate this result in
the form of the following theorem.
Theorem 6.11. For a given p=1,2,3,… we have:
DetQp=(1)p .
(6.56)
Let us calculate the determinants of the inverse matrices of the kind Q p−1 . For
this purpose we consider the wellknown correlation, which connects the “di
rect” and “inverse” matrices:
Q p−1 × Q 1p = I ,
(6.57)
where I is the identity matrix with
DetI=1.
(6.58)
Applying to (6.57) the wellknown rule given by Theorem 6.3 for the deter
minant of the product of two matrices and taking into consideration (6.58), we
can write:
Alexey Stakhov
THE MATHEMATICS OF HARMONY
332
Det Q p−1 × Det Q 1p = Det I = 1.
(6.59)
In accordance with (6.56) the determinant of the matrix Q 1p takes one of the
two values, 1 or 1; it follows from (6.59) that
Det Q p−1 = Det Q 1p = 1.
(6.60)
Let us calculate the determinant of the matrix Q pn using (6.1):
n
Det Q pn = Det Q p ,
(6.61)
(
)
where n=0,±1,±2,±3,… .
Taking into consideration the result of Theorem 6.11, we can formulate
the following theorem.
Theorem 6.12. For given p=1,2,3,… and n=0,±1,±2,±3,…, the determi
nant of the matrix Q pn is given by the following expression:
(6.62)
Det Q pn = ( −1) .
It is clear that Theorem 6.12 is a generalization of the wellknown Theorem
6.6 for the Qmatrix that corresponds to the case p=1.
Note that Theorems 6.8 and 6.12 are a source for new results in the field of
Fibonacci number theory [16].
For example, consider the matrix Q2n for the case p=2
pn
 F2 (n + 1) F2 (n) F2 (n − 1) 
Q2n =  F2 (n − 1) F2 (n − 2) F2 (n − 3) 


 F2 (n) F2 (n − 1) F2 (n − 2) 
(6.63)
where n=0,±1,±2,±3,… .
By calculating the determinant of the matrix (6.63) and by using the
identity (6.62), we can write the case for p=2:
Det Q2n = 1.
(6.64)
It is easy to prove the following theorem.
Theorem 6.13. For p=2 and n=0,±1,±2,±3,…, we have the following iden
tity connecting the adjacent Fibonacci 2numbers:
Det Q2n = F2 ( n + 1)  F2 ( n − 2) F2 ( n − 2 ) − F2 ( n − 1) F2 ( n − 3 )
+ F2 ( n )  F2 ( n ) F2 ( n − 3 ) − F2 ( n − 1) F2 ( n − 2 ) 
+ F2 ( n − 1)  F2 ( n − 1) F2 ( n − 1) − F2 ( n ) F2 ( n − 2 ) = 1.
(6.65)
It is clear that Theorem 6.8 and 6.12 give a theoretically infinite number of
correlations similar to (6.6) and (6.65). If we remember that Fibonacci pnum
bers were obtained in the study of the diagonal sums of Pascal’s triangle, we can
assert that the identities similar to (6.6) and (6.65) express some unusual math
ematical properties of Pascal’s triangle!
Chapter 6
Fibonacci and Golden Matrices
333
6.5.The “Direct” and “Inverse” Fibonacci Matrices
Again, let us consider Table 6.1, which gives the “direct” and “inverse” Q
matrices. By comparing the “direct” (Qn) and “inverse (Qn) Fibonacci Q
matrices, it is easy to find a very simple method that allows us to obtain the
“inverse” matrix Qn from its “direct” matrix Qn.
In fact, if the power n of the “direct” matrix Qn given by (6.4) is even
(n=2k), then for obtaining its inverse matrix Qn it is simply necessary to
interchange the places of the diagonal elements Fn+1 and Fn1 in (6.4) and to
change the sign of the diagonal elements Fn. This means that for the case n=2k
the “inverse” matrix Qn has the following form:
− F2 k 
F
Q −2 k =  2 k −1
.
(6.66)
 − F2 k F2 k +1 
In order to obtain the “inverse” matrix Qn from the “direct” matrix Qn given by
(6.4) for the case n=2k+1, it is necessary to interchange the places of the diagonal
elements Fn+1 and Fn1 in (6.4) and give them the opposite sign, that is:
F2 k +1 
 −F
Q −2 k −1 =  2 k
.
(6.67)
 F2 k +1 − F2 k +2 
Another way to obtain the matrices Qn follows directly from the expression
(6.4). Here we can represent two Fibonacci series Fn+1 and Fn1 shifted by one
number with respect to the other (Table 6.2).
If we select the num
Table 6.2. Fibonacci series Fn+1 and Fn1
ber n=1 in the first row
n
5 4 3 2 1 0 −1 −2 −3 −4 5 −6
of Table 6.2 and then se
lect the four Fibonacci
1 −1 2 3 5
Fn +1 8 5 3 2 1 1 0
numbers in the lower
Fn
5 3 2 1 1 0 1 −1 2 −3 5 8
two rows under the num
ber 1 and to the right with respect to it, then the totality of these Fibonacci
numbers form the Qmatrix. The Qmatrix is singled out in bold type in Table
6.2. If we move in Table 6.2 to the left with respect to the Qmatrix, then we
obtain the matrices Q2, Q3,…, Qn, respectively. If we move in Table 6.2 to the
right with respect to the Qmatrix, then we obtain the matrices Q0, Q1,…, Qn,
respectively. Also the Fibonacci matrices Q5 and the “inverse” to it, Fibonac
ci matrix Q5 are singled out in bold type in Table 6.2. Note that the matrix Q0
is an identity matrix.
This principle of the Qmatrices design can be used for the general case of
Qpmatrices. The analysis of the matrix (6.30) for the case p=2 shows that all
THE MATHEMATICS OF HARMONY
Alexey Stakhov
334
matrices of the kind Q2n can be obtained from (6.30), provided we represent
three series of the adjacent Fibonacci 2numbers F2(n+1), F2(n1), F2(n), shift
ed with respect to each other as is shown in Table 6.3.
Table 6.3. Fibonacci 2numbers F2(n+1), F2(n1), F2(n)
n
5
4
3
2
1
0
−1
−2
−3
−4
5
−6
−7
F2 ( n + 1)
4
3
2
1
1
1
0
0
1
0
1
1
1
2
1
1
1
0
0
1
0
−1
1
1
2
0
3
2
1
1
1
0
0
1
0
−1
1
1
2
F2 ( n − 1)
F2 ( n )
Select the number n=1 in the first row of Table 6.3 and then select the known
Fibonacci 2numbers in the lower rows with respect to the number 1 as is shown
in Table 6.3 (they are singled out by bold type). It is clear that the totality of
singled out Fibonacci 2numbers form the Q2matrix.
If we move to the left in Table 6.3 with respect to the Q2matrix, then we
obtain the matrices Q22 , Q23 , ..., Q2n , respectively. If we move to the right in
Table 6.3 with respect to the Q 2matrix, then we obtain the matrices
Q20 = I , Q2−1, ..., Q2− n , respectively.
By analogy, using (6.30), we can construct the table of the Fibonacci
3numbers that give the matrices of the kind Q3n (see Table 6.4).
Table 6.4. Fibonacci 3numbers F3(n+1), F3(n2),F3(n1), F3(n)
n
5
4
3
2
1
0
−1
−2
−3
−4
−5
−6
−7
F3 ( n + 1)
3
2
1
1
1
1
0
0
0
1
0
0
−1
1
1
1
0
0
0
1
0
0
−1
1
0
1
1
1
1
1
0
0
0
1
0
0
−1
1
0
2
1
1
1
1
0
0
0
1
0
0
−1
1
F3 ( n − 2)
F3 ( n − 1)
F3 ( n )
6.6. Fibonacci GmMatrices
6.6.1. A Definition of the Fibonacci GmMatrix
In Chapter 4 we introduced the generalized Fibonacci mnumbers
given by the recursive relation
F m(n+2)=mFm(n+1)+Fm(n)
at the seeds
(6.68)
Chapter 6
Fibonacci and Golden Matrices
335
Fm(0)=0, Fm(1)=1,
(6.69)
where m>0 is a given real number and n=0,±1,±2,±3,… .
Similar to the Fibonacci Qmatrix (6.2), which is a generating matrix for
the classical Fibonacci numbers, we can introduce the Gmmatrix [118] that
is a generating matrix for the Fibonacci mnumbers given by the recursive
relation (6.68) at the seeds (6.69).
Gmmatrix
( )
Gm = m 1 .
1 0
(6.70)
Note that the determinant of the Gmmatrix (6.70) is equal to −1:
Det Gm =m×0 −1×1=−1.
(6.71)
The following theorem gives a connection of the Gmmatrix (6.70) to the
Fibonacci mnumbers given by (6.68) and (6.69) .
Theorem 6.14. For a given integer n=0,±1,±2,±3,…, the nth power of
the Gmmatrix is given by
 F (n + 1) Fm (n) 
Gmn =  m
,
 Fm (n) Fm (n − 1) 
where Fm(n−1), Fm(n), Fm(n+1) are the Fibonacci mnumbers.
Proof. We use mathematical induction. Clearly, for n=1
(6.72)
 F (2) Fm (1) 
Gm1 =  m
(6.73)
.
 Fm (1) Fm (0) 
By using the seeds (6.69) and the recursive relation (6.68), we can write:
Fm(0)=0, Fm(1)=1, Fm(2)=mFm(1)+ Fm(0)=m.
It follows from (6.73) and (6.74) that
( )
Gm1 = m 1 .
1 0
(6.74)
(6.75)
The basis of the induction is therefore proved.
Suppose that for a given integer k our inductive hypothesis is the following:
 F k +1 Fmk 
Gmk =  m k
.
k −1 
 Fm Fm 
We can then write:
Fm (k)   m 1 
 F (k + 1)
Gmk+1 = Gmk × Gm =  m
×

Fm (k − 1)   1 0 
 Fm (k)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
336
 mF ( k + 1) + Fm ( k ) Fm ( k + 1)   Fm ( k + 2 ) Fm ( k + 1) 
= m
=
.
 mF ( k ) + F ( k − 1)
Fm ( k )   Fm ( k + 1)
Fm ( k ) 
m
 m
Thus, the theorem is proved.
6.6.2. Determinants of the GmMatrices
The next theorem gives a formula for the determinant of the matrix (6.72).
Theorem 6.15. For a given integer n=0,±1,±2,±3,…, we have:
Det Gmn = (−1) .
n
(6.76)
Proof. By using the property (6.1) and by taking into consideration (6.71),
we can write:
Det Gmn = ( Det Gm ) = (−1) .
Thus, the theorem is proved.
n
n
6.6.3. GmMatrix and Cassini Formula for the Fibonacci mNumbers
If we calculate the determinant of the matrix (6.72) and take into consid
eration (6.76), we obtain the following identity:
(6.77)
Det Gmn = Fm (n + 1)× Fm (n −1)− Fm2 (n) = (−1) .
Note that the identity (6.77) is one of the most important identities for
the Fibonacci mnumbers. It is clear that the identity (6.77) is a generaliza
tion of the wellknown Cassini formula (6.6).
n
6.6.4. Some Properties of the GmMatrices
Theorem 6.16.
Gmn = mGmn−1 + Gmn−2 .
(6.78)
Proof. By using the recursive relation (6.68), we can represent the matrix
(6.72) as follows:
mFm (n − 1) + Fm (n − 2) 
 mFm (n) + Fm (n − 1)
Gmn = 

 mFm (n − 1) + Fm (n − 2) mFm (n − 2) + Fm (n − 3) 
 F (n)
Fm ( n − 1)   Fm ( n − 1) Fm ( n − 2 ) 
+
= mGmn−1 + Gmn−2 .
= m m
 F ( n − 1) F ( n − 2 )   F ( n − 2 ) F ( n − 3 ) 
m
m
  m

 m
Thus, the theorem is proved.
Chapter 6
Fibonacci and Golden Matrices
337
We can also represent the recursive relation (6.78) in the following form:
Gmn−2 = Gmn − Gmn−1 .
(6.79)
Based on the recursive relations (6.78) and (6.79), we can construct the se
quences of the Gmmatrices (6.72) similar to Table 6.1. Note that for the case
m=1 the matrices G1n coincide with the matrices Qn, that is, Table 6.1 gives a
sequence of the matrices G1n .
Consider the case m=2. Remember that for this case a sequence of the
Fibonacci mnumbers Fm=2(n) looks similar to Table 6.5.
Table 6.5. A sequence of the Fibonacci mnumbers for the case m=2
n
6
5
4
3
2
1
0
−1
−2
−3
−4
−5
−6
F2 ( n )
70
29
12
5
2
1
0
1
−2
5
−12
29
−70
Let us construct a sequence of the matrices G2n . For the case n=0, we define
the matrix Gmn as follows:
( )
Gm0 = 1 0 .
0 1
(6.80)
Using the recursive relation (6.78) and taking into consideration the seeds (6.80)
and (6.75), we can construct the matrices G22 , G23 , G24 and so on as follows:
( )( ) ( )
G = 2 ( 5 2) + ( 2 1) = (12 5)
21
10
5 2
G = 2 (12 5) + (5 2) = ( 29 12) .
5 2
21
12 5
G22 = 2 2 1 + 1 0 = 5 2
10
0 1
21
(6.81)
3
2
(6.82)
4
2
(6.83)
Using the recursive relation (6.79) and taking into consideration the seeds
(6.80) and (6.75), we can construct the matrices G2−1 , G2−2 , G2−3 and so on as
follows:
( ) ( )( )
G = ( 1 0 ) − 2 ( 0 1 ) = ( 1 −2 )
01
1 −2
−2 5
G = ( 0 1 ) − 2 ( 1 −2) = ( −2 5 ) .
−2 5
1 −2
5 −12
G2−1 = 2 1 − 2 1 0 = 0 1
10
0 1
1 −2
(6.84)
−2
2
(6.85)
−3
2
(6.86)
A sequence of the matrices G2n is represented in Table 6.6.
THE MATHEMATICS OF HARMONY
Alexey Stakhov
338
Table 6.6. A sequence of the matrices G2n
n
0
1
2 1
1 0
0 1
1 −2
2
3
12 5
5 2
5
−2
5 −12
( ) ( ) ( ) (
( ) ( ) ( ) (
1
0
1
0
n
2
G
G2− n
0
1
0
1
5
2
1
−2
2
1
−2
5
) (
) (
4
) (
) (
29
12
5
−12
12
5
−12
29
5
70
29
−12
29
)
29
12
29
−70
)
Consider the case m=3. Remember that for the case m=3 a sequence of the
Fibonacci mnumbers Fm(n) looks similar to Table 6.7.
Table 6.7. A sequence of the Fibonacci mnumbers for the case m=3
n
6
5
4
3
2
1
0
−1
−2
−3
−4
−5
−6
F3 ( n )
360
109
33
10
3
1
0
1
−3
10
−33
109
−360
Using the recursive formulas (6.78) and (6.79) at the seeds (6.75) and (6.80)
for the case m=3 we can construct the matrices G3n (see Table 6.8).
Table 6.8. A sequence of the matrices G3n
n
0
1
3 1
1 0
0 1
1 −3
2
10 3
3 1
1 −3
−3 10
( ) ( ) (
( ) ( ) (
1
0
1
0
n
3
G
−n
3
G
0
1
0
1
3
33 10
10 3
−3 10
10 −33
) (
) (
4
109 33
33 10
10 −33
−33 109
) (
) (
5
360 109
109 33
−33 109
109 −360
) (
) (
)
)
It is easy to verify that all square matrices of the kind Gmn in Table 6.6 and
Table 6.8 possess one surprising property: all their determinants are equal to +1
(for the even powers n) or (−1) (for the odd powers n). In fact, the determinant
( )
(
) is equal to (−33)×(−360)−109×109=−1.
29 12
of the matrix G2 = 12 5 is equal to 29×5−12×12=1 and the determinant of the
4
−33 109
matrix G3 = 109 −360
Now, let us consider a general case of m. Remember that the number m takes
its value from the range of positive real numbers, for example, m can take the
following values: m = 2, π, e (the base of natural logarithms) and so on. Using
the recursive relation (6.78) and taking into consideration the seeds (6.75) and
(6.80), we can construct the matrices Gm2 , Gm3 , Gm4 and so on for a general case of
m:
−5
m 1   1 0   m2 + 1 m 
Gm2 = m 
+
=

 1 0 0 1  m 1 
(6.87)
 2
 m 1   m 3 + 2m m2 + 1
Gm3 = m  m + 1 m  + 
= 2
m 
 m 1   1 0  m + 1
(6.88)
Chapter 6
Fibonacci and Golden Matrices
339
 m 3 + 2m m2 + 1  m2 + 1 m   m 4 + 3m2 m 3 + 2m 
Gm4 = m  2
+
=
.
(6.89)
m   m 1   m 3 + 2m m2 + 1 
 m +1
Using the recursive relation (6.79) and taking into consideration the seeds
(6.75) and (6.80), we can construct the matrices Gm−1 , Gm−2 , Gm−3 and so on for
another general case of m:
( ) ( ) (
Gm−1 = m 1 − m 1 0 = 0 1
1 0
0 1
1 −m
)
(6.90)
 1 −m 
Gm−2 = 
2

 −m m + 1
(6.91)
 −m
m2 + 1 
Gm−3 =  2
3
.
 m + 1 − m − 2m 
(6.92)
6.6.5. The Inverse Matrices Gmn
Again, let us consider Tables 6.6 and 6.8. They set the “direct” and “inverse”
Gmmatrices. Compare the “direct” and the “inverse” Gmmatrices, Gmn and Gm−n .
It is easy to find a very simple method to obtain the “inverse” matrix Gm−n from
its “direct” matrix Gmn .
In fact, if the power n of the “direct” matrix Gmn given by (6.72) is
even (n=2k), then for obtaining its inverse matrix Gm−n it is necessary
to interchange the places of the diagonal elements F m(n+1) and F m(n
1) in (6.72) and to take the diagonal elements F m (n) in (6.72) with the
opposite sign. This means that for the case n=2k the “inverse” matrix
Gm−2 k has the following form:
 F (2k − 1) − Fm (2k) 
Gm−2 k =  m
(6.93)
.
 − Fm (2k) Fm (2k + 1) 
To obtain the “inverse” matrix Gm−n from the “direct” matrix Gmn given by
(6.72) for the case n = 2k + 1, it is necessary to interchange the places of the
diagonal elements Fm(n+1) and Fm(n1) in (6.72) and to take them with the
opposite sign, that is:
 − F (2k − 1) Fm (2k) 
Gm−2 k −1 =  m
(6.94)
.
 Fm (2k) − Fm (2k + 1) 
Another way of obtaining the matrices Gmn follows directly from the ex
pression (6.72). Let us consider the case m=2. In order to get the matrices
G2n (n=0,±1,±2,±3,…) we have to represent two Fibonacci series F2(n+1)
and F2(n) shifted by one number with respect to the other (Table 6.9).
Alexey Stakhov
THE MATHEMATICS OF HARMONY
340
Table 6.9. The shifted Fibonacci series F2(n+1) and F2(n)
n
6
5 4 3 2 1 0 −1 −2 −3 −4 5 −6
F2 ( n + 1) 169 70 29 12 5 2 1 0 1 −2 5 12 29
F2 ( n )
70 29 12 5 2 1 0 1 −2 5 −12 29 70
If we select the number n=1 in the first row of Table 6.9 and then select four
Fibonacci numbers in the lower two rows under the number 1 and to the right
with respect to it, then a totality of these Fibonacci numbers build up the Gm
matrix (6.72). The Gmmatrix is singled out by bold type in Table 6.9. If we move
in Table 6.9 to the left with respect to the Gmmatrix, then we obtain the matri
ces G22 , G23 , G24 and so on. If we move in Table 6.9 to the
right with respect to the G mmatrix, we then obtain the matrices
G20 = I , G2−1 , G2−2 and so on. Note that the matrix G25 and the “inverse” to it ma
trix G2−5 are singled out by bold type in Table 6.9. Note that the matrix G20 = 1 0
01
is an identity matrix.
( )
This method of obtaining the matrices Gmn can be used for any arbitrary m.
6.7. Fibonacci Qp,mMatrices
6.7.1. A Definition of the Qp,mMatrix
In order to define the Qp,mmatrix, let us return back to the recursive rela
tion (4.295) at the seeds (4.296) that generate the Fibonacci (p,m)numbers. By
analogy with the Qpmatrix (6.28), which is a generating matrix for the Fibonac
ci pnumbers Fp(n), we introduce the Qp,mmatrix (6.95), which is a generating
matrix for the recursive relation (4.295) at the seeds (4.296). Given integer p>0
and real number m>0, the generating matrix for the Fibonacci (p, m)numbers is
 m 1 0 0 " 0 0
 0 0 1 0 " 0 0
 0 0 0 1 " 0 0
Q p ,m =  # # # # # # #  .
(6.95)
 0 0 0 0 " 1 0
 0 0 0 0 " 0 1
 1 0 0 0 " 0 0


Note that the expression (6.95) defines an infinite number of square matri
ces because every positive real number m generates its own generating matrix of
Chapter 6
Fibonacci and Golden Matrices
341
the kind (6.95). The Qp,mmatrix (6.95) is a square (p+1)×(p+1)matrix. It con
sists of the identity (p×p)matrix bordered by the last row consisting of 0’s and
the leading 1, and the first column consisting of 0’s embraced by the upper ele
ment of m and the lower element of 1. We can list the Qp,mmatrices for the case
p=1, 2, 3, 4 as follows:
 m 1 0 0 0
 m 1 0 0
0 0100
 m 1 0
0
0
1
0
1
m
; Q =  0 0 0 1 0.
Q1,m =
= Gm ; Q 2,m =  0 0 1  ; Q 3,m = 


4
,
m
1 0
 1 0 0
 0 0 0 1
0 0 0 0 1  (6.96)



1
0
0
0


 1 0 0 0 0
Note that the Qp,mmatrix (6.95) is a wide generalization of the Qmatrix
(6.2) (p=1, m=1), the Qpmatrix (6.28) (m=1) and Gmmatrix (6.70) (p=1).
( )
6.7.2. The Main Theorems for the Powers of the Qp,mMatrices
Now, let us raise the Qp,mmatrix (6.95) to the nth power and find the
analytical expression for matrix Q pn,m . We have the following theorem for the
Qp,mmatrices.
Theorem 6.17. For a given integer p=1,2,3,…, we have:
Fp , m (n) " Fp , m (n − p + 2) Fp, m (n − p + 1) 
 Fp ,m (n + 1)
 F (n − p + 1) F (n − p) " F (n − 2 p + 2) F (n − 2 p + 1) 
p,m
p,m
p,m
 p,m

n
Q p,m = 
#
#
#
#
#
,
Fp , m (n − p − 1) 
 Fp ,m (n − 1) Fp ,m (n − 2) " Fp , m (n − p)
 F ( n)
Fp , m (n − 1) " Fp , m (n − p + 1)
Fp , m (n − p) 
p,m

(6.97)
where Fp,m(n) is the Fibonacci (p,m)number, and n=0,±1,±2,±3,… .
Proof. For a given p we can use an induction method.
(a) The basis of the induction. First, we will prove the basis of the induc
tion, that is, we prove that the statement (6.97) is valid for the case n=1. It is
clear that for the case n=1 the matrix (6.97) takes the following form:
Fp ,m (1) " Fp ,m (− p + 3) Fp ,m (− p + 2) 
 Fp,m (2)
 Fp ,m (− p + 2) Fp ,m (− p + 1) " Fp ,m (−2 p + 3) Fp,m (−2 p + 2) 

.
#
#
#
#
#


Fp ,m (−1) " Fp ,m (− p + 1)
F p , m ( − p) 
 Fp,m (0)
 F (1)
Fp ,m (0) " Fp ,m (− p + 2) Fp ,m (− p + 1) 
p ,m

(6.98)
Taking into consideration the values of the initial terms of the Fibonacci
(p,m)numbers given by (4.296) and their mathematical properties given by
(4.297), (4.299) and (4.300), we can write:
Fp,m(2)=m, Fp,m(2)=1 and Fp,m(p)=1.
THE MATHEMATICS OF HARMONY
Alexey Stakhov
342
Note that all the remaining entries of the matrix (6.98) are equal to 0. This
means that the matrix (6.98) coincides with the Qp,mmatrix given by (6.95).
The basis of the induction is proved.
(b) Inductive hypothesis. Suppose that the statement (6.97) is valid for an
arbitrary n and prove it for n +1. For this purpose we consider the product of
the matrices
Q pn,+m1 = Q pn,m ×Q p,m
m 1 0 0 " 0 0
F ( n) " F ( n − p + 2) F ( n − p + 1)   0 0 1 0 " 0 0 
 F ( n + 1)
 F ( n − p + 1) F ( n − p) " F ( n − 2 p + 2) F ( n − 2 p + 1)   0 0 0 1 " 0 0 
 ×  # # # # % # # .
=
#
#
#
#
#
(6.99)
 F ( n − 1) F (n − 2) " F (n − p) F ( n − p − 1)   0 0 0 0 " 1 0 
 F (n)
 0 0 0 0 " 0 1
F ( n − 1) " F ( n − p + 1)
F (n − p )  



 1 0 0 0 " 0 0
Let us consider the matrix Q pn+1 = (aij ) . In order to calculate the entry
a11, we multiply term by term the first row of the matrix Q pn,m by the first
column of the matrix Qp,m as follows:
p,m
p,m
p,m
p,m
p,m
p,m
p,m
p,m
p,m
p,m
p,m
p,m
p,m
p,m
p,m
p,m
a 11=mFp,m(n+1)+Fp,m(np+1).
(6.100)
According to the recursive relation (4.295) we have: a11=Fp,m(n+2).
Continuing the process of the matrix multiplication for the example
(6.99), we obtain the following entries of the first row of the matrix Q pn,+1
m :
Fp,m(n+2), Fp,m(n+1), …, Fp,m(np+3), Fp,m(np+2).
(6.101)
Now, let us compare the first row of the matrix Q pn,m given by (6.97) with
the first row of the matrix Q pn,+1
m given by (6.101). We can see that all argu
ments of the Fibonacci (p,m)numbers in the sequence (6.101) differ by +1
from the arguments of the corresponding Fibonacci (p,m)numbers, which
build up the first row of the matrix (6.97). By analogy, we can show that this
rule is valid for all entries of the matrix Q pn,+1
m that are formed from the matrix
Q pn,m by means of its multiplication by the matrix Qp,m in accordance with
(6.99). This consideration proves the validity of Theorem 6.17.
Also, it is easy to prove the following identity for the product of these
Q pn,m matrices:
Q pn,+m1 = Q pn,m ×Q p,m = Q p,m ×Q pn,m .
Using (6.102), we can prove the following theorem.
Theorem 6.18.
(6.102)
Q pn,m ×Q pk ,m = Q pk ,m ×Q pn,m = Q pn,+mk ,
(6.103)
where n and k are integers.
Chapter 6
Fibonacci and Golden Matrices
343
Decomposing each entry of the matrix (6.97), that is, the corresponding Fi
bonacci (p,m)number and taking into consideration the recursive relation
(4.295), we obtain the following result given by Theorem 6.19.
Theorem 6.19.
Q pn,m = mQ pn−,m1 + Q pn−,mp−1 .
(6.104)
The recursive relation (6.104) can also be represented in the following form:
Q pn−,mp−1 = Q pn,m −Q pn−,m1 .
(6.105)
Using (6.104) and (6.105), we can construct different sequences of the
Q pn,m matrices. For example, for the case p=1 the sequences of the Q1,nm matri
ces for the cases m=2 and m=3 are given by Tables 6.6 and 6.8, respectively.
6.8. Determinants of the Qp,mMatrices and their Powers
6.8.1. Determinant of the Qp,mMatrix
For the case p=1 the Qp,mmatrix is reduced to the Gmmatrix, that is:
( )
Q1,m = m 1 = Gm .
1 0
(6.106)
According to (6.71) we have:
DetQ1,m=−1.
Now, let us calculate the determinant of the Q2,mmatrix
(6.107)
 m 1 0
Q2,m =  0 0 1  .
(6.108)
 1 0 0


By comparing the Q2,mmatrix (6.108) with the Q1,mmatrix (6.106), we
can see that the Q2,mmatrix (6.108) is reduced to the Q1,mmatrix (6.106) if
we strike out the 2nd row and the 3rd column from the Q2,mmatrix (6.108).
Note that the sum of these numbers 2+3=5. According to the matrix theory
[157] we have:
Det Q2,m=1×(1)5Det Q1,m=1×(1)×(1)=1.
(6.109)
Now, let us calculate the determinant of the Q3,mmatrix
 m 1 0 0
Q3,m =  0 0 1 0  .
0 0 0 1
 1 0 0 0


(6.110)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
344
Comparing the matrix (6.110) with the matrix (6.108), we can see
that the Q3,mmatrix (6.110) is reduced to the Q2,mmatrix (6.108) if we
strike out the 3rd row and the 4th column (3+4=7) in the matrix (6.110).
This means [157] that
Det Q3,m=1×(−1)7×Det Q2,m=1×(−1)×1=−1.
By analogy, we can show that
(6.111)
(6.112)
Det Q4,m=1×(−1)9×Det Q3,m=1×(−1)×(−1)=1.
Continuing this process, that is, generating the formulas similar to (6.109),
(6.111), and (6.112), we can write the following recursive relation, which
connects the determinants of the adjacent matrices Qp,m and Qp1,m :
(6.113)
Det Qp,m=1×(−1)2p+1×Det Qp1,m .
It is clear that Det Qp,m is equal to 1 for the even values of p=2k and to (1)
for the odd values of p=2k +1. We can formulate this result in the form of the
following theorem.
Theorem 6.20. For a given p=1,2,3,…, we have:
(6.114)
Det Qp,m=(−1)p.
Now, let us calculate the determinants of the inverse matrices of the kind
−1
Q p,m . For this purpose we consider this wellknown correlation connecting
the “direct” and “inverse” matrices:
Q −p,1m ×Q 1p,m = I ,
(6.115)
where I is the identity matrix with Det I=1.
Applying to (6.115) the wellknown rule that is given by Theorem 6.3, we
can write:
Det Q −p,1m × Det Q 1p,m = Det I .
(6.116)
In accordance with (6.114) the determinant of the matrix Det Qp,m takes
one of the two values, 1 or 1. It follows from (6.116) that
Det Q −p,1m = Det Q p,m .
(6.117)
6.8.2. Determinant of the Matrix Q pn,m
Let us calculate the determinant of the matrix Q pn,m given by (6.97). Us
ing the property (6.1), we can write:
Det Q pn,m = (Q p,m ) ,
(6.118)
where n=0,±1,±2,±3,… .
Taking into consideration the result of Theorem 6.20, we can formulate
the following theorem.
n
Chapter 6
Fibonacci and Golden Matrices
345
Theorem 6.21. For given p=1,2,3,… and n=0,±1,±2,±3,… the determi
nant of the matrix Q pn,m is given by the following formula:
Det Q pn,m = (−1)
pm
.
(6.119)
6.9. The Golden QMatrices
6.9.1. A Definition of the Golden Matrices
We can represent the matrix (6.4) in the form of two matrices that are given
for the even (n=2k) and odd (n=2k+1) values of n:
F 
F
Q 2 k =  2 k +1 2 k 
(6.120)
 F2 k F2 k −1 
F 
F
Q 2 k +1 =  2 k +2 2 k +1  .
(6.121)
F
 2 k +1 F2 k 
In Chapter 5 we introduced the socalled Symmetric Hyperbolic Fibonacci
and Lucas Functions (5.57)(5.60) that are connected with the Fibonacci and
Lucas numbers by the simple correlations (5.61). Consider once again the
symmetric hyperbolic Fibonacci functions:
Symmetric hyperbolic Fibonacci sine
τ x − τ− x
sFs ( x ) =
(6.122)
5
Symmetric hyperbolic Fibonacci cosine
τ x + τ− x
cFs ( x ) =
(6.123)
5
Remember that the symmetric hyperbolic Fibonacci functions (6.122) and
(6.123) are connected by the following surprising identities:
 sFs ( x ) − cFs ( x − 1) cFs ( x − 1) = −1;
2
(6.124)
2
cFs ( x ) − sFs ( x − 1) sFs ( x − 1) = −1.
Note that the identities (6.124) are a generalization of the Cassini formu
la for the continuous domain.
Remember that the Fibonacci numbers Fn are connected with the func
tions (6.122) and (6.123) by the simple correlation:
Alexey Stakhov
THE MATHEMATICS OF HARMONY
346
for n = 2k
 sFs ( n ) ,
Fn = 
(6.125)
cFs ( n ) , for n = 2k + 1.
We can use the correlation (6.125) to represent the matrices (6.120) and
(6.121) in terms of the symmetric hyperbolic Fibonacci functions (6.122)
and (6.123):
sFs(2k) 
 cFs(2k + 1)
Q 2k = 

cFs(2k − 1) 
 sFs(2k)
(6.126)
 sFs(2k + 2) cFs(2k + 1) 
,
(6.127)
sFs(2k) 
 cFs(2k + 1)
where k is a discrete variable, k=0,±1,±2,±3,… .
If we exchange the discrete variable k in the matrices (6.126) and (6.127)
for the continuous variable x, we obtain two unusual matrices that are func
tions of the continuous variable x:
Q 2k +1 = 
sFs(2 x ) 
 cFs(2 x + 1)
Q 2x = 

cFs(2 x − 1) 
 sFs(2 x )
(6.128)
 sFs(2 x + 2)
cFs(2 x + 1) 
.
(6.129)
sFs(2 x ) 
 cFs(2 x + 1)
It is clear that the matrices (6.128) and (6.129) are a generalization of the Q
matrix (6.4) for the continuous domain. They have a number of unique mathe
matical properties. For example, for x=1/4 the matrix (6.128) takes the follow
ing form:

3
1 
 cFs   sFs   
1
2
2 
.
Q2 = Q =
(6.130)

 1 
1
 sFs   cFs  −  
 2 
2

It is difficult to even imagine what a “square root of the Qmatrix” means.
However, such an amazing “Fibonacci fantasy” follows directly from (6.130).
Q 2 x +1 = 
6.9.2. The Inverse Golden Matrices
In the above we introduced the “inverse” Fibonacci Qmatrices given
by (6.66) and (6.67). We can represent the inverse of matrices (6.66) and
(6.67) in terms of the symmetric hyperbolic Fibonacci functions (6.122)
and (6.123) as follows:
Chapter 6
Fibonacci and Golden Matrices
347
 cFs(2k − 1) − sFs(2k) 
Q −2 k = 

 − sFs(2k) cFs(2k + 1) 
(6.131)
 − sFs(2k)
cFs(2k + 1) 
,
(6.132)
 cFs(2k + 1) − sFs(2k + 2) 
where k is a discrete variable, k=0,±1,±2,±3,… .
If we exchange the discrete variable k in the matrices (6.131) and (6.132)
for the continuous variable x, we then obtain the following matrices that are
functions of the continuous variable x:
Q −2k−1 = 
 cFs(2 x − 1) − sFs(2 x ) 
Q −2 x = 

 − sFs(2 x ) cFs(2 x + 1) 
(6.133)
 − sFs(2 x )
cFs(2 x + 1) 
.
(6.134)
 cFs(2 x + 1) − sFs(2 x + 2) 
It is easy to prove that the matrices (6.133) and (6.134) are the inverse of the
matrices (6.128) and (6.129), respectively, that is,
Q −2 x −1 = 
Q2x× Q2x =I and Q2x+1× Q2x1 =I ,
where I is an identity matrix.
6.9.3. Determinants of the Golden Matrices
Now, let us calculate the determinants of the matrices (6.128) and
(6.129):
Det Q 2 x = cFs ( 2 x + 1) cFs ( 2 x − 1) −  sFs ( 2 x ) 
2
(6.135)
Det Q 2 x +1 = cFs ( 2 x + 2 ) cFs ( 2 x ) −  sFs ( 2x + 1)  .
2
(6.136)
Compare formulas (6.135) and (6.136) with the identities (5.124) for the
symmetric hyperbolic Fibonacci functions. As the identities (6.124) are valid
for all values of the variable x, in particular, for the value of 2x, the next
identities follow from (6.135), (6.134) and (6.124):
DetQ2x=1
(6.137)
DetQ2x+1=1
(6.138)
Note that the unusual identities (6.137) and (6.138) are a generalization
of the “Cassini formula” for the continuous domain.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
348
6.10. The Golden GmMatrices
6.10.1. A Definition of the Golden GmMatrices
The Fibonacci Gmmatrices (6.72) are a source for the wide generalization of
“golden” matrices introduced above. We can represent the Gmmatrix (6.72) in
the form of two matrices that are given for the even (n=2k) and odd (n=2k+1)
values of n:
Fm (2k) 
 F (2k + 1)
Gm2 k =  m

Fm (2k − 1) 
 Fm (2k)
 F (2k + 2) Fm (2k + 1) 
Gm2 k +1 =  m
.
Fm (2k) 
 Fm (2k + 1)
Let us consider the hyperbolic Fibonacci and Lucas mfunctions.
Hyperbolic Fibonacci mfunctions
sFm ( x ) =
cFm ( x ) =
Φ mx − Φ m− x
4 + m2
Φ mx + Φ m− x
4 + m2
x
−x

2 
 m + 4 + m2  
m
m
+
+
4

=
 − 
 
2
2
4 + m 2  
 
 

(6.139)
(6.140)
1
(6.141)
−x
x

2 
 m + 4 + m2  
4
m
m
+
+

=
 + 
  . (6.142)
2
2
4 + m2  
 
 

1
The Fibonacci mnumbers are determined identically by the hyperbolic Fi
bonacci and Lucas mfunctions as follows:
for n = 2k
 sF ( n ) ,
Fm ( n ) =  m
cFm ( n ) , for n = 2k + 1.
(6.143)
As is shown in Chapter 5, the hyperbolic Fibonacci mfunctions (6.141)
and (6.142) possess the following unique properties:
 sFm ( x ) − cFm ( x + 1) cFm ( x − 1) = −1
(6.144)
cFm ( x ) − sFm ( x + 1) sFm ( x − 1) = 1.
(6.145)
2
2
Using (6.143), we can represent the matrices (6.139) and (6.140) in terms
of the hyperbolic Fibonacci mfunctions (6.122) and (6.123) as follows:
Chapter 6
 cFm (2k + 1)
Gm2k = 
 sFm (2k)
Fibonacci and Golden Matrices
349
sFm (2k) 

cFm (2k − 1) 
(6.146)
 sF (2k + 2) cFm (2k + 1) 
Gm2 k +1 =  m
,
(6.147)
sFm (2k) 
 cFm (2k + 1)
where k is a discrete variable, k=0,±1,±2,±3,… .
If we exchange the discrete variable k in the matrices (6.146) and (6.147)
for the continuous variable x, we then obtain two unusual matrices that are
functions of the continuous variable x:
sFm (2 x ) 
 cF (2 x + 1)
Gm2 x =  m

cFm (2 x − 1) 
 sFm (2 x )
(6.148)
 sF (2 x + 2) cFm (2 x + 1) 
Gm2 x +1 =  m
.
(6.149)
sFm (2 x ) 
 cFm (2 x + 1)
Note that the “golden” Gmmatrices given by (6.148) and (6.149) provide a
wide generalization of the “golden” Qmatrices given by (6.128) and (6.129).
The “golden” Qmatrices (6.128) and (6.129) are partial cases of the matrices
(6.148) and (6.149) for the case m = 1, that is,
G12 x = Q 2 x and G12 x +1 = Q 2 x +1 .
(6.150)
6.10.2. Inverse Golden Gmmatrices
We can represent the inverse Gmmatrices (6.93) and (6.94) in terms of
the hyperbolic Fibonacci mfunctions (5.105) and (5.106), that is,
 cF (2k − 1) − sFm (2k) 
Gm−2 k =  m

 − sFm (2k) cFm (2k + 1) 
(6.151)
cFm (2k + 1) 
 − sFm (2k)
Gm−2 k −1 = 
,
(6.152)
 cFm (2k + 1) − sFm (2k + 2) 
where k is a discrete variable, k=0,±1,±2,±3,… .
If we exchange the discrete variable k in the matrices (6.151) and (6.152)
for the continuous variable x, we then obtain the following matrices, which
are functions of the continuous variable x:
 cF (2 x − 1) − sFm (2 x ) 
Gm−2 x =  m

 − sFm (2 x ) cFm (2 x + 1) 
(6.153)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
350
cFm (2 x + 1) 
 − sFm (2 x )
Gm−2 x −1 = 
.
(6.154)
 cFm (2 x + 1) − sFm (2 x + 2) 
It is easy to prove that the matrices (6.153) and (6.154) are the inverse of the
matrices (6.148) and (6.149), respectively, that is,
Gm2 x ×Gm−2 x = I and Gm2 x +1 ×Gm−2 x −1 = I ,
where I is an identity matrix.
6.10.3. Determinants of the Golden GmMatrices
Calculate the determinants of the “golden” mmatrices (6.148) and (6.149):
Det Gm2 x = cFm ( 2 x + 1) × cFm ( 2 x − 1) −  sFm ( 2 x )
2
(6.155)
Det Gm2 x +1 = sFm ( 2 x + 2 ) × sFm ( 2 x ) − cFm ( 2 x + 1) .
2
(6.156)
Let us compare the formulas (6.155) and (6.156) with the identities (5.136)
and (5.137) for the hyperbolic Fibonacci mfunctions. As the identities (5.136)
and (5.137) are valid for all values of the variable x, in particular, for the value of
2x, the following identities follow from this consideration:
Det Gm2 x = 1
(6.157)
(6.158)
Det Gm2 x +1 = −1.
Note that the unusual identities (6.157) and (6.158) are further generaliza
tions of the Cassini formula for the continuous domain.
6.11. The Golden Genomatrices by Sergey Petoukhov
6.11.1. The Genetic Code
The discovery of the genetic code, which is general for all living organ
isms from a bacterium to a man, led us to the development of the informa
tional point of view for living organisms. As it is emphasized in [59], “from
this point of view all organisms are informational essences. They exist
because they get hereditary information from ancestors and they live to
transfer the informational genetic code to descendants. Given such an
approach we may treat all other physical and chemical mechanisms pre
sented in living organisms as auxiliary, helping promote the realization of
this basic, informational problem.”
Chapter 6
Fibonacci and Golden Matrices
351
The basis for the hereditary information language is amazingly simple. For
recording genetic information in the ribonucleic acids (RNA) of any organism,
the “alphabet” that consists of the four “letters” of nitrogenous bases, is used:
Adenine (A), Cytosine (C), Guanine (G), Uracil (U) [in DNA instead of
Uracil(U) the related Thymine (T) is used].
The genetic information transferred by molecules of heredity (DNA and
RNA) defines the primary structure of the protein of the living organism. Each
coded protein represents a chain that consists of 20 kinds of amino acids. The
block formed from three adjacent nitrogen bases is known as a Triplet. It is
possible to make 43=64 triplets from the fourletter alphabet. The genetic code
is called a Degenerate one because 64 triplets code only 20 amino acids. If any
protein chain contains n amino acids, then the sequence of triplets that corre
sponds to it contains 3n nitrogen bases in the DNA molecule or, in other words,
is given by a 3ntriplet. Protein chains usually contain hundreds of amino acids
and are accordingly represented by rather long polytriplets.
Recently the Russian researcher Sergey Petoukhov made an original dis
covery in the genetic code [59]. This discovery shows the fundamental role of
the golden ratio in the genetic code. It provides further evidence that the
golden ratio underlies all Living Nature!
6.11.2. Symbolic Genomatrices
Petoukhov’s basic idea [59] consists of the representation of the genetic
code in matrix form. A square (2×2)matrix P is an elementary matrix that is
used for the representation of a system of four nitrogen bases (“letters”) of
the genetic alphabet:
( )
P= C A .
U G
(6.159)
Sergey Petoukhov further suggests that one may represent a family of
genetic codes of identical length in the form of a corresponding family of
matrices P(n) that are tensor (Kronecker) degrees of the initial matrix (6.159).
The matrices P(n) are named Symbolic Genomatrices. For a large enough n
the given family of symbolic genomatrices P(n) represents all systems of ge
netic polyplets including Monoplets of the genetic alphabet (6.159) and
Triplets (6.161) that are used for coding amino acids.
In each of the four quadrants of the genomatrix P(n) we can see all nplets
that begin from one of the letters C, A, U, G. If we ignore the first letter of the
nplets, we can see that every quadrant of the matrix P(n) reproduces the
matrix P(n1) of the preceding generation. On the other hand, every matrix
Alexey Stakhov
THE MATHEMATICS OF HARMONY
352
P(n) builds up a quadrant of the matrix P(n1)of the next generation. Thus, a genom
atrix of every new generation contains in itself in latent form all the information
of the preceding generations. It is pertinent here to compare the genomatrices
P(n) and the Qpmatrices (6.28). Examples of the matrices P(2) and P(3) are repre
sented below:
 CC CA AC AA 
( 2)
P = P ⊗ P =  CU CG AU AG 
UC UA GC GA
(6.160)
UU UG GU GG 


 CCC CCA CAC CAA ACC ACA AAC AAA 
 CCU CCG CAU CAG ACU ACG AAU AAG 
 CUC CUA CGC CGA AUC AUA AGC AGA 
 CUU CUG CGU CGG AUU AUG AGU AGG 
( 3)
P =P⊗P⊗P =
.
 UCC UCA UAC
C UAA GCC GCA GAC GAA 
(6.161)
 UCU UCG UAU UAG GCU GCG GAU GAG 
 UUC UUA UGC UGA GUC GUA
A GGC GGA 
 UUU UUG UGU UGG GUU GUG GGU GGG 


We can see that the matrices (6.159)(6.161) possess the properties sim
ilar to the matrix (6.28) because any Qpmatrix (6.28) comprises the informa
tion about all the previous matrices Qp1, Qp2,…, Q1, Q0. However, on the other
hand, the Q pmatrix (6.28) is contained in all the next matrices
Qp+1, Qp+2, Qp+3,… .
6.12.3. Numerical Genomatrices
If we substitute some numerical parameter for every symbol of nitro
gen bases in the symbolic genomatrix, we obtain a Numerical Genomatrix.
To form such numerical parameters, Petoukhov suggests that one uses the
numerical parameters of the complementary hydrogen relation for the ni
trogen bases of the genetic code. We are talking about the two or three
hydrogen relations that connect complementary pairs of nitrogen bases in
molecules of heredity. For the bases C and G the number of such nitrogen
relations is equal to 3; however, for A and U it is equal to 2. Petoukhov
suggested the following rule for obtaining the numerical genomatrix from
the corresponding symbolic genomatrix.
Petoukhov’s rule 1. To obtain the numerical genomatrix from the corre
sponding symbolic genomatrix, it is necessary to substitute every polyplet
for the product of the numbers of hydrogen relations of its nitrogen bases,
namely A=U=2 and C=G=3.
For example, triplet CGA in the octet matrix (6.164) is replaced by the
product 3×3×2=18.
Chapter 6
Fibonacci and Golden Matrices
353
As a result of such substituting, the symbolic genomatrices (6.159)(6.161)
are converted, respectively, into the following numerical genomatrices Pmult:
( )
(1)
Pmult
= 3 2
2 3
 9 6 6 4
(2)
Pmult =  6 9 4 6 
6 4 96
 4 6 6 9


(6.162)
(6.163)
 27 18 18 12 18 12 12 8 = 125 
 18 27 12 18 12 18 8 12 = 125 
 18 12 27 18 12 8 18 12 = 125 
 12 18 18 27 8 12 12 18 = 125 
(3) 
Pmult
= 18 12 12 8 27 18 18 12 = 125 
 12 18 8 12 18 27 12 18 = 125  .
(6.164)
 12 8 18 12 18 12 27 18 = 125 
 8 12 12 18 12 18 18 27 = 125 
 125 125 125 125 125 125 125 125 1000 


It is easy to find a number of interesting properties of the numerical genom
atrices (6.162) (6.164). First of all, all numerical genomatrices (6.162) (6.164) are symmetric with respect to both diagonals and therefore are called
bisymmetric [141]. Further, the sum of numbers of each line and each column
of the matrices (6.162) (6.164) are equal to 5, 52=25, and 53=125, respec
tively, and the total sums of numbers in the matrices (6.162) (6.164) are
equal to 10, 102=100, and 103=1000, respectively. It is proven [141] that any
( n ) possesses similar properties because each such
numerical genomatrix Pmult
matrix is bisymmetric. Thus, the sum of numbers of each line and each column
is equal to 5n, and the total sum of numbers in the matrix is equal to 10n.
Already these surprising properties of Petoukhov’s numerical genomatrices
create the impression of “magic.” But Petoukhov’s discovery [59] of an im
probable connection of these numerical genomatrices to the golden mean is
absolutely astounding!
6.12.4. The Golden Genomatrices
Next let us consider the numerical genomatrix (6.162) that corresponds
to the simplest symbolic genomatrix (6.159).
Consider a square matrix Φ(1), where the numbers τ = 1 + 5 2 (the
−1
golden mean) and τ = −1 + 5 2 are its elements:
−1


Φ (1) =  τ−1 τ  .
τ
τ


(
)
(
)
(6.165)
THE MATHEMATICS OF HARMONY
Alexey Stakhov
354
If we square the matrix (6.165), we obtain the following matrix:
−1
−1
−2
2
2
2 .
Φ (1)  =  τ−1 τ  ×  τ−1 τ  =  τ + τ






 τ
τ  τ
τ   2
τ −2 + τ2 
(6.166)
Now, let us recall the Lucas numbers Ln (1, 3, 4, 7, 11, 18, ...) and the Binet
formula for Lucas numbers. We can represent the Binet formula for Lucas
numbers as follows:
n
Ln = τn + (−1) τ−n ,
(6.167)
where n=0,±1,±2,±3,… .
For the case n=2 the identity (6.167) takes the following form:
(6.168)
τ 2+τ 2=L 2=3.
Taking into consideration the identity (6.168), we can represent the ma
trix (6.166) as follows:
( ) = (32 32).
Φ( )
1
2
(6.169)
If we compare the matrices (6.162), (6.165), and (6.169), we can see a
deep mathematical connection between the numerical genomatrix (6.162)
and the “golden” genomatrix (6.165), because after squaring the “golden”
genomatrix (6.165) we come to the numerical genomatrix (6.169).
( n)
Petoukhov proved [59] that every numerical genomatrix of the kind Pmult
has the “golden” genomatrix Φ(n), which after its squaring coincides with the
initial numerical genomatrix. Petoukhov suggested the following rule for
obtaining the “golden” genomatrix from the corresponding symbolic genom
atrix.
Petoukhov’s rule 2. To obtain the “golden” genomatrix from the corre
sponding symbolic genomatrix, it is necessary to substitute every polyplet
for the product of the following values of its letters: C=G=τ, A=U=τ1.
If we apply this rule to the symbolic genomatrix (6.160), we obtain the
following “golden” genomatrix:
 τ 2 τ 0 τ 0 τ −2 
 0 2 −2 0 
2
Φ ( ) =  τ0 τ−2 τ 2 τ0  .
 τ−2 τ 0 τ0 τ2 
τ τ τ τ 
If we square the matrix (6.170), we obtain:
τ +τ +τ +τ
τ + τ + τ + τ
(2)
 Φ  = 
τ +τ +τ +τ

 τ +τ +τ +τ
2
4
0
0
−4
2
2
−2
−2
τ +τ +τ +τ
2
−2
2
−2
τ +τ +τ +τ
0
0
0
τ +τ +τ +τ
0
(6.170)
2
2
−2
−2
0
4
−4
0
τ +τ +τ +τ
0
0
0
0
τ +τ +τ +τ
−2
−2
τ +τ +τ +τ
2
2
−2
2
−2
0
0
0
0
0
−4
4
0
τ +τ +τ +τ
−2
−2
2
2
τ
2
τ +τ +τ +τ
τ +τ +τ +τ
0
0
0
τ +τ +τ +τ

0
τ +τ +τ +τ 
−2
−2
−4
2
−2
2
.

+τ +τ +τ 
−2
0
2
0
2
−4
(6.171)
Chapter 6
Fibonacci and Golden Matrices
355
If we use the Binet formula (6.167), we can write the following identities:
(6.172)
τ2+τ2=3 and τ4+τ4=7.
Taking into consideration the identities (6.172), we can represent the ma
trix (6.171) as follows:
 9 6 6 4
 Φ (2)  =  6 9 4 6  .
2

6 4 9 6
 4 6 6 9


After comparison of (6.163) and (6.173) we can write:

2
(2)
Φ (2)  = Pmult
,


(6.173)
(6.174)
that is, the “golden” genomatrix (6.170), which was obtained from the sym
bolic genomatrix (6.160) by using Petoukhov’s rule 2, is converted after its
squaring into the numerical genomatrix (6.173), which was obtained from
the symbolic genomatrix (6.160) by using Petoukhov’s rule 1.
Now, let us consider the “golden” genomatrix Φ(3), which can be obtained
from the symbolic genomatrix (6.163) by using Petoukhov’s rule 2:
 τ3 τ1 τ1 τ −1 τ1 τ −1 τ −1 τ −3 
 1 3 − 1 1 − 1 1 −3 −1 
 τ1 τ−1 τ 3 τ1 τ −1 τ−3 τ 1 τ −1 
τ τ 
τ τ τ τ
τ τ
−1
1
1
3
−3
−1
−1

τ
τ
τ
τ
τ
τ
τ
τ1 
3
Φ ( ) =  1 −1 −1 −3 3 1 1 −1  .
τ τ τ τ 
τ τ τ τ
 τ −1 τ1 τ −3 τ −1 τ1 τ3 τ −1 τ1 
 −1 −3 1 −1 1 −1 3 1 
τ τ
τ τ
τ τ 
τ τ
 τ −3 τ −1 τ −1 τ1 τ −1 τ1 τ1 τ3 


(6.175)
If we square the matrix (6.175), then after simple transformation we obtain:
2
( 3)
Φ (2)  = Pmult
.


(6.176)
It appears that a similar regularity is valid for any numerical genomatrix
( n)
. This means that Sergey Petoukhov proved the following important
Pmult
proposition named Petoukhov’s Discovery.
Petoukhov’s discovery.
Let A (adenine), C (cytosine), G (guanine), and U (uracil) be nitrogen
bases (“letters”) of the genetic alphabet that build up the initial symbolic
matrix P = UC GA . Let P(n) be a symbolic genomatrix formed by means of a
tensor (Kronecker) raising of the initial symbolic matrix P to the nth power
( )
Alexey Stakhov
THE MATHEMATICS OF HARMONY
356
and let polyplets, which are built up from the “letters” A, C, G and U, be the
( n)
elements of the symbolic matrix P(n). If we build a numerical genomatrix Pmult
(n)
from the symbolic genomatrix P by substituting for each polyplet of the
matrix P(n) the products of the numbers of hydrogen relations of its nitrogen
bases according to the rule: A=U=2 and C=G=3, and if we build a “golden”
genomatrix Φ(n) from the symbolic genomatrix P(n) by substituting for each
polyplet of the symbolic matrix P(n) the products of the following values of its
“letters” according to the rule: C=G=τ and A=U=τ1 , where τ = 1 + 5 2 is
the golden mean, then there is the following fundamental relation between
( n) and the “golden” genomatrix Φ(n):
the numerical genomatrix Pmult
(
)
2
(n)
Φ ( n )  = Pmult
.


This discovery of the relation between the golden mean and the genetic
code allowed Petoukhov to give a new “matrixgenetic” definition based upon
the golden mean.
Petoukhov’s definition.
The golden mean and its inverse number (τ and τ1) are the only matrix
elements of the bisymmetric matrix Φ that is a root square of such bisym
metric numerical genomatrix Рmult of the second order whose elements are
genetic numbers of hydrogen relations (C=G=3, A=U=2).
The “golden” genomatrices of Sergey Petoukhov are the most surprising
application of the matrix approach to the golden section theory. Petoukhov’s
discovery [59] shows a fundamental role of the “golden mean” in the genetic
code. This discovery gives further evidence that the golden mean underlies
all Living Nature! It is difficult to estimate the full impact of Petoukhov’s
discovery for the development of modern science. It is clear that this scientif
ic discovery is of equal importance to the discovery of the genetic code!
We would like to end this chapter by quoting from Petoukhov’s paper
[59] emphasizing the importance of the matrix approach in Fibonacci num
ber and golden mean theory:
“The above formulated proposition about the matrix definition and es
sence of the golden section gives the possibility of considering all this mate
rial and its informative interpretation from the fundamentally new, matrix
point of view. The author believes that many realizations of the golden sec
tion in both Organic and Inorganic Nature are connected precisely to the
matrix essence and representation of the golden section. The mathematics of
the golden matrices is a new branch of mathematics, which studies recur
rence relations between the series of the golden matrices and also models
natural systems and processes with their help.”
Chapter 6
Fibonacci and Golden Matrices
357
6.12. Conclusion
1. Thus in this chapter we have developed a theory of the special class of
square matrices that have exceptional mathematical properties. The begin
ning of this theory is connected with the name of the American mathemati
cian Verner Hoggatt – founder of the Fibonacci Association. The theory of
the Fibonacci Qmatrix was first stated in Hoggatt’s book [16]. Although
the name of the Qmatrix, which is a generating matrix for the classical Fi
bonacci numbers, was introduced before Verner E. Hoggatt, it was from
Hoggatt’s research that the idea of the Qmatrix “caught on like wildfire
among Fibonacci enthusiasts. Numerous papers have appeared in ‘The Fi
bonacci Quarterly’ authored by Hoggatt and/or his students and other col
laborators where the Qmatrix method became a central tool in the analysis
of Fibonacci properties” [158].
2. Alexey Stakhov in his 1999 article [103] generalized the concept of the
Qmatrix and introduced the Generalized Fibonacci Qpmatrix, which is a gen
erating matrix for the Fibonacci pnumbers. In 2006 Stakhov introduced
[118] the concept of a Gmmatrix, which is a generating matrix for the Fi
bonacci mnumbers. E. Gokcen Kocer, Naim Tuglu and Alexey Stakhov in
troduced [154] the concept of the Qp,mmatrix, which is a generating matrix
for the Fibonacci (p,m)numbers. The “golden” Q and Gmmatrices are based
on the symmetric hyperbolic Fibonacci functions and the hyperbolic Fibonac
ci mfunctions. A general property of all the above matrices is the fact that
their determinants are equal to +1 or 1. It is clear that a theory of similar
matrices is of general mathematical interest.
3. However, the “golden” genomatrices of Sergey Petoukhov [59] are the
most surprising application of the matrix approach to the golden mean theo
ry. Petoukhov’s discovery [59] demonstrates the fundamental role of the
“golden mean” in the genetic code. This discovery gives further evidence that
the golden mean underlies all Living Nature! Petoukhov’s discovery may
actually be scientifically equal in importance to the discovery of the genetic
code itself!
This page intentionally left blank
Part III
Part III.
Application
in Computer Science
Alexey Stakhov
THE MATHEMATICS OF HARMONY
360
Chapter 7
Algorithmic Measurement Theory
7.1. The Role of Measurement in the History of Science
7.1.1. What is a Measurement?
In the Great Soviet Encyclopedia we can find the following definition of
the measurement notion:
“Measurement is an operation, by means of which the ratio of one magni
tude to another homogeneous magnitude is determined; the number, which rep
resents this ratio, refers to the numerical value of the subject magnitude.”
A measurement is an important method of quantitative cognition of the
objective world. The eminent Russian scientist Dmitry Mendeleev, the
founder of the Periodic System of Chemical Elements and Father of Russian
Metrology, maintained: “Science begins with a measurement. Exact science is
inconceivable without a measure.”
7.1.2. The “Differentiation Principle” of “Measurement Science”
The concept of a Magnitude and its Measurement belongs among the basic
concepts of science. The theoretical study of this concept began developing in
ancient Greece. During the 20th century the representatives of the different
scientific disciplines, philosophy, physics, mathematics, information theory,
psychology, economy, and so on, paid focused attention to this concept. That
is why, the problem of the definition of the subject, contents and the position
of measurement science in the system of contemporary sciences is of great im
portance. In this Section we make a methodological analysis of the ideas, prin
ciples, and scientific theories that can be united under the general name Sci
ence of Measurement or Measurement Science.
The existence of various directions of measurement study in modern
science is a reflection of dialectic process of the Measurement Science dif
Chapter 7
Algorithmic Measurement Theory
361
ferentiation as a major principle of its development. Measurement as a
method of quantitative reflection of the objective properties of the Uni
verse is a dialectically manysided concept. Every exact science studies
measurement from its own specific point of view. It selects the measure
ment property that is most important for that given science, and a study of
this measurement property results in a special measurement theory. For
example, in quantum physics the most essential property of quantum mea
surement is the interaction between microobject and the macromeasure
ment device. This problem underlies QuantumMechanical Measurement
Theory [159]. In sociology, psychology, systems theory, and economics the
measurement is reduced to the choice of the type of scale, to which the
measurable magnitude can be referred, and therefore the Scale Problem un
derlies the Psychological Measurement Theory [160]. For technical and phys
ical measurement, the main problem of measurement is the choice of the
System of Physical Units and the decrease of Measurement Errors. Study of
these aspects of measurement resulted in the creation of Metrology the
science of technical and physical measurements [161]. Measurement er
rors can be considered to be “random noises” in the “measurement chan
nel;” this idea underlies the Information Measurement Theory [162]. The
study of measurement, as some method or algorithm to get a numerical
result, then results in the Algorithmic Measurement Theory [20, 21].
7.1.3. Applied and Fundamental Theories of Measurement
As we mentioned above, the general measurement theory concerns all
branches of science and technology. On the one hand, Metrology is an applied
science that concerns the Engineering Sciences and Applied Physics. On the
other hand, measurement concerns the fundamental problems of science, in
particular, Mathematics and Theoretical Physics. Therefore, at all stages of
the development of science and technology, two levels or aspects of measure
ment study exist: Applied and Fundamental.
The Fundamental Level assumes a study of measurement as a fundamental
problem that is a source of the development of the exact sciences (in particu
lar, physics, mathematics, nonphysical exact sciences); on this level the prob
lem consists of revealing the most general properties and laws of measurement
as the method of quantitative cognition of the objective world. Discovery
of these measurement laws had a substantial influence on the development
of the exact sciences. The proof of the existence of Incommensurable Seg
ments made by the Pythagoreans is an example of just such a unique math
Alexey Stakhov
THE MATHEMATICS OF HARMONY
362
ematical discovery. This discovery determined the development of mathe
matics over millennia because it resulted in Irrational Numbers, one of the
most important mathematical concepts. Heisenberg’s “Uncertainty Rela
tions” that underlie quantum physics fundamentally limit the exactness of
quantummechanical measurement and may be considered to be the out
standing physical idea in the field of the measurement.
The Applied Level assumes the study of measurement from the point
of view of the practical, applied problems that appear in technology and
applied physics. The problem of the creation of Systems of Measurement
Units (metric system, Gauss’ absolute system, etc.) that runs throughout
the history of science and technology is the most typical example of the
applied measurement problem.
7.1.4. What is the Fundamental Distinction between Physics and
Mathematics?
Physics and mathematics education programs do not, as a rule, give much
attention to this important question. Its answer lies, for example, in the works
of the famous physicist Brilluen [163]. By pondering mathematical theorems
and physical theories, Brilluen defines the essence of the distinction as follows.
While mathematics begins with definitions of dimensionless points, infinitely
thin curves, and continuous spacetime, modern physics denies any real sense to
such definitions. Brilluen noted that “for physicists the irrational numbers do
not have any significance. It is assumed that the irrational numbers need to be
defined with infinite precision because the exactness of say one in a hundred, one
in a million or even one in a billion is insufficient for their definition. For the
physical experiment, this does not make any sense; we can measure within the
fifth or tenth digits of a decimal fraction, but there is no experiment, which would
give twenty decimal digits. There is no sense in asking, whether the speed of
light is a rational number, whether the ratio M/m (the ratio of the proton weight
M to the electron weight m) is rational. All irrational numbers, like π, e, 2 ,
etc., only follow from abstract mathematical definitions. However, as we have
already said, mathematics is an aimless art. No physical objects can show real
geometrical properties in those borders, which are assumed by mathematicians;
we can even name such an approach, “wishful thinking” [163].
Thus, according to Brilluen, a feature of the “physical” approach to the study
of the objective world consists in the postulation of some limit of “definiteness”
of the “physical” magnitude that is completely ignored in the purely mathemat
ical approach to the definition of the concept of magnitude.
Chapter 7
Algorithmic Measurement Theory
363
7.1.5. A Thesis about Inevitability of Measurement Errors
Hence, according to Brilluen, the distinction between physical and mathe
matical approaches to measurement theory consists of different attitudes toward
the Thesis about the Inevitability of Measurement Errors. The physical approach
recognizes this thesis as the main postulate of the physical measurement theory.
This axiom has an empirical, practical substantiation. The Russian mechanic
and academician Krylov noted that “every measurement of any magnitude al
ways has some error. Clearly, the less the error, the more exact the measure
ment, but the practice of measurement shows that it is impossible to avoid error
entirely. This is confirmed by the fact that when iterating the same measure
ment many times by the same device we will get measurement results, which are
different between themselves” [161].
This axiom finds its theoretical acknowledgement in the fact of the existence
of some limit of definiteness (negentropy) of physical magnitude, which is known
in quantum mechanics as Heisenberg’s Uncertainty Relation, in wave mechanics
as a Relation of TimeandFrequency Uncertainty, and on the molecular level is
given by the laws of thermodynamics (Brilluen’s Negentropy Principle of Infor
mation). The existence of the limit of definiteness of physical magnitude and, as
a consequence, the impossibility of the “absolutely exact” comparison (distinc
tion) of the sizes of two physical magnitudes, which differ from each other in the
size of the definiteness limit, is the basic thesis of “physical” measurements. Thus,
the concept of “magnitude” in physics differs from a similar concept in mathe
matics, where we adopt the opposite thesis about the possibility of the “abso
lutely exact” comparison of two magnitudes independently of their sizes (com
parison axioms of the mathematical theory of magnitudes).
7.1.6. Purposes of the “Physical” Measurement Theory
If we follow Brilluen’s approach, we can divide the “fundamental” part of
measurement theory into the Physical and Mathematical measurement theo
ries. This approach allows us to formulate the basic purpose of the “physical”
measurement theory which follows from the basic thesis about the inevitability
of measurement errors. In quantum physics [159], the main purpose of mea
surement theory is the study of the interaction of the quantummechanical ob
ject with the macrodevice used for the quantummechanical measurement. In
metrology and the applied physics, the main purpose of the measurement theo
ry is to study and model the errors of the physical measurements and in recent
decades to study the sensitivity threshold of measurement devices. Of course,
Alexey Stakhov
THE MATHEMATICS OF HARMONY
364
the doctrine about physical magnitudes and units [161] always was an impor
tant branch of the “physical” measurement theory.
Study of the character of physical measurement errors resulted in the formu
lation of two empirical axioms, which underlie the theory of random errors [161]:
1. Randomness’ axioms: for the very big number of measurement re
sults, the random errors, equal in terms of absolute value, but varying by
sign, meet equally often; the number of the negative errors is equal to the
number of the positive errors.
2. Distribution’s axiom: smaller errors meet more often, than bigger; very
big errors do not meet.
These empirical axioms resulted in the wellknown Normal Law of Probabil
ity Distribution (Gauss’ law):
2
x
−
1
ϕ(x ) =
e 2 σ2 ,
σ 2π
2
where σ is a dispersion of probabilities; x is current value of error.
In the opinion of Russian mathematical authority Tutubalin [164], “the ‘Nor
mal Law’ is some kind of “miracle” of probability theory, without which this
theory almost would not have the original contents ....”
The correctness of the above axioms and the “normal law,” which follows
from them, is confirmed by the fact that all conclusions, based on them, are
always consistent with experience. However, the nonstrict character of the
formulated axioms always causes a certain dissatisfaction with Gauss’ law.
This situation is expressed by one witty mathematician who sarcastically
noted that “the experimenters believe in Gauss’ law, by relying on the proofs
of mathematicians, and the mathematicians believe in this law, by relying on
the experimental verification.”
7.2. Mathematical Measurement Theory
7.2.1. Evolution of the Measurement Concept in Mathematics
A measurement problem plays the same exceptional role in mathematics, as in
other areas of science, particularly in technology, physics, and other exact sciences.
The Russian mathematician and academician Kolmogorov in the foreword to the
Russian edition of Lebesgue’s The Measurement of the Magnitudes [3] expressed
his outlook on this problem in the following words: “Where is the main interest of
Chapter 7
Algorithmic Measurement Theory
365
Lebesgue’s book? It seems to me, it is in the following: mathematicians have a
propensity already owning the finished mathematical theory to be ashamed of its
origin. In comparison with the crystal clearness of the development of a theory,
beginning from its already prepared basic concepts and assumptions, it seems dirty
and unpleasant to delve into the origin of these basic concepts and assumptions.
The entire edifice of school algebra and all mathematical analysis can be construct
ed solely on the concept of real number without any reference to the measurement
of concrete magnitudes (lengths, areas, time intervals, etc.). Therefore, on the
different steps of education with a different degree of boldness, one and the same
tendency emerges: as soon as it is possible to be done with the introduction of
number concept and immediately to start discussing only numbers and relations
between them. This is the tendency Lebesgue protests against this!”
Let us track now the evolution of the measurement concept in mathemat
ics. It is well attested that, the measurement rules used by the Egyptian land
surveyors were the first “measurement theory.” The ancient Greeks testify
that geometry was obliged by its origin (and the name) to those measurement
rules, that is, to the “measurement problem.” However, in Ancient Greece the
separation of the measurement problems into two parts began – the applied
problems related to “logistics,” which was at that time referred to as a set of
certain rules for the applied measurements and calculations; and fundamental
problems related to geometry and number theory. These latter fundamental
measurement problems became the main problems of Greek mathematics.
In the Greek period, a science of measurement was developing primarily as a
mathematical theory. During this period, the Greek mathematicians made the
following mathematical discoveries: Incommensurable Segments, Eudoxus’ Meth
od of Exhaustion and the Measurement Axiom. These outstanding mathematical
discoveries later became the sources of number theory, integration and differen
tiation and other mathematical theories. These mathematical discoveries had a
direct impact upon the “measurement problem.” It gave the Bulgarian mathema
tician and academician Iliev the impetus to proclaim that “during the first epoch
of mathematics development, from antiquity to the discovery of differential and
integral calculus, investigating first of all the measurement problems, mathemat
ics had created Euclidean geometry and number theory” [5].
7.2.2. Incommensurable Line Segments
From high school on we believe in the strictness and stability of mathemat
ics. Therefore, for some of us it is a big surprise that in the process of its develop
ment, mathematics was experiencing different crises. Moreover, it may be a big
Alexey Stakhov
THE MATHEMATICS OF HARMONY
366
ger surprise for others that, since the beginning of the 20th century, modern
mathematics is in a deep crisis, and contemporary mathematicians cannot see
any way out of this crisis yet.
The first crisis in the foundations of mathematics was connected directly to
the “measurement problem.” The early Pythagorean mathematics was based upon
the socalled Commensurability Principle. Let us recall, that in mathematics,
two nonzero real numbers a and b are said to be Commensurable if and only if
a/b is a rational number. According to the commensurability principle, any two
geometric magnitudes Q and V have a common measure, that is, both magni
tudes are divisible by it. Thus, their ratio can be expressed as the ratio of two
mutually prime natural numbers m and n:
Q m
= .
(7.1)
V
n
According to the main Pythagorean philosophical doctrine that Everything
is a number, all geometrical magnitudes can be expressed in the form (7.1), that
is, their ratio is always rational. This means that geometry is reduced to a num
ber theory because according to (7.1) all geometric relations can be expressed
by rational numbers.
Let us examine the ratio of the diagonal b and the side a of the square
(Fig. 7.1). According to Pythagorean Theorem
(7.2)
b2=2a2.
It follows from (7.2) that the ratio of the diagonal b to the side a of the
square (Fig. 7.1) is equal
(7.3)
b / a = 2.
Suppose that the diagonal b and the side a of the square are commensura
ble. This means that we can represent their ratio in the form 2 = m n , where
m and n are the mutually prime natural numbers. Then m2=2n2. Hence, it
follows from here that the number m2 is an even number. However, if a square
of a number is even, this means that the number m is also an even number. As the
numbers m and n are mutually prime natural numbers, then the number n is an
odd number, according to the definition (7.1). Howev
a
er, if m is even number, the number m2 can be divided
by 4, and hence, n2 is an even number. Thus, n is also an
b
b
even number. However, n cannot be an even and odd
2
a
number simultaneously! This contradiction shows that
our premise about the commensurability of the diago
Figure 7.1. Incommensura nal and the side of the square is wrong and therefore the
number 2 is irrational.
ble line segments.
Chapter 7
Algorithmic Measurement Theory
367
The discovery of incommensurability staggered the Pythagoreans and caused
the first crisis in the basis of mathematics, as this discovery refuted the initial
Pythagorean doctrine about commensurability of all geometric magnitudes. A
discovery of irrational numbers originated a complicated mathematical notion,
which did not have direct connection with human experience. According to the
legend, Pythagoras made a “hecatomb,” that is, sacrificed one hundred oxen to
the gods in honor of this discovery. However, according to the Brewer’s Dictio
nary, “He sacrificed to the gods millet and honeycomb, but not animals. He
forbade his disciples to sacrifice oxen.” In any case, it was a worthwhile sacrifice
because this discovery became the turning point in the development of mathe
matics. It ruined the former system created by the Pythagoreans (the commen
surability principle) and became a source of new and remarkable theories. The
importance of this discovery may be compared with the discovery of nonEuclid
ean geometry in the 19th century or to the theory of relativity at the beginning
of 20th century. Similar to these theories, the problem of incommensurable line
segments was wellknown among educated people. Plato and Aristotle were also
addressing the problem of incommensurability in their works.
7.2.3. EudoxusArchimedes and Cantor’s Axioms
To overcome the first crisis in the mathematics foundations, the Great
Greek mathematician Eudoxus developed the method of exhaustion and
created a new theory of magnitudes.
Eudoxus of Cnidus (c.410/408 – 355/347 BC) was a Greek astronomer,
mathematician, physicist, scholar and friend of Plato. Unfortunately, all of
his own works were lost, therefore we obtained information about him from oth
er sources, such as Aratus’ poem on astronomy. Archytas from Athens was his
mathematics teacher. He is famous due to the introduction of the astronomical
globe, and his early contributions to the explanation of the planetary motion.
Eudoxus developed the method of exhaustion and used it for the creation
of the theory of irrationals. Later this method was used masterfully by
Archimedes. The essence of Eudoxus’ method of exhaustion can be explained
by the following practical example. If we have a barrel of beer and a beer mug,
then this beer in the barrel will eventually be exhausted, even though the beer
barrel is enormous and the beer mug very small.
Eudoxus’ theory of incommensurability (see Book V of Euclid’s Ele
ments) could be considered one of the greatest achievements of mathe
matics and in general coincides with the modern theory of irrational num
bers suggested by Dedekind in 1872.
THE MATHEMATICS OF HARMONY
Alexey Stakhov
368
The measurement theory of geometric magnitudes dates back to the incom
mensurable line segments. It is based on the group of socalled continuity axi
oms, which comprise both EudoxusArchimedes’ axiom and Cantor’s axiom (or
Dedekind’s axiom).
EudoxusArchimedes’ axiom (the axiom of measurement). For any
two segments A and B it is possible to specify such natural number n, which
always results in the following nonequality:
nB>A.
(7.4)
In the 19th century, Dedekind and then Cantor made a final attempt to
create a general theory of real numbers. For this purpose, they introduced
the additional axioms into the group of the continuity axioms. For instance, let
us consider Cantor’s Axiom.
Cantor’s continuity axiom (Cantor’s principle of nested segments). If
an infinite sequence of segments is given on a straight line A 0B0, A 1B 1,
A2B2,…,AnBn, …, such that each next segment is nested within the preceding
one, and the length of the segments tends to zero, then there exists a unique
point which belongs to all the segments.
A
A0
A1
A2
An
Bn
B2
B1
B0
C
B
B
B
B
a)
B
B
b)
Figure 7.2. The continuity axioms: (a) EudoxusArchimedus’ axiom
(b) Cantor’s axiom
The main result of the mathematical measurement theory that is based on
the continuity axioms is a proof of the existence and uniqueness of the solution q
of the Basic Measurement Equality:
Q=qV,
(7.5)
where V is a measurement unit, Q is a measurable segment, and q is any real
number named a Result of Measurement.
The idea of the proof of the equality (7.5) consists of the following. Using Eu
doxusArchimedes’ axiom and certain rules called a Measurement Algorithm, we
can form from the measurement unit V some sequence of the nested segments,
which can be compared with the measurable segment Q during the process of mea
surement. If we direct this process ad infinitum, then according to Cantor’s axiom
for the given line segments Q and V we always can find such nested segments,
which coincides with the measurable segment Q. It follows from this proof that
the idea about a measurement is one of a process running during an infinite time.
Chapter 7
Algorithmic Measurement Theory
369
It is difficult to imagine that the continuity axioms and the mathematical
measurement theory are the result of a more than 2000year period in the devel
opment of mathematics. The continuity axioms and basic measurement equa
tion (7.5) comprise in themselves a number of the important mathematical ideas
that underlie different branches of mathematics.
It is necessary to note that the measurement axiom expressed by (7.4) is a
reflection of Eudoxus’ method of exhaustion in modern mathematics. The axiom
generalizes mankind’s millennia of experience regarding the measurement of dis
tances, areas and time intervals. The simplest measurement algorithm of the line
segment A by using the line segment B less than A underlies this axiom. This
algorithm consists of the successive comparison of the line segment B+B+…+B
with the measurable line segment A. During the measurement process we count
a number of the line segments B that are laid out on the measurable line segment
A. This measurement algorithm is called a Counting Algorithm.
The counting algorithm is the source of the various fundamental notions of
arithmetic and number theory, in particular, the notions of Natural Number, the
Prime and Composite numbers, as well as the notion of Multiplication, Division, etc.
In this connection the Euclidean definition of prime (the “first” number) and com
posite numbers (“the first number is measured only by 1,” “the composite number
is measured by some number”) is of great interest. The measurement axiom origi
nates a Divisibility Theorem, which plays a fundamental role in number theory. A
Theory of Divisibility and a Comparison Theory are based on the Divisibility Theo
rem. Thus, the simplest measurement algorithm, a Counting Algorithm, originates
natural numbers and all concepts and theories connected with them.
7.2.4. A Contradiction in the Continuity Axioms
The idea about measurement as a process running during infinite time is a
brilliant example of the Cantorian style of mathematical thinking based on
the concept of Actual Infinity. However, this concept was subjected to sharp
criticism from the side of the representatives of constructive mathematics.
For the explanation of the distinctions between different interpretations
of infinity used in EudoxusArchimedes’ and Cantor’s axioms, once again we
return back to them. In Cantor’s axiom all infinite sets of the nested segments
are considered as the set given by all its objects simultaneously (Cantor’s actu
al infinity). EudoxusArchimedes’ axiom, which has an “empirical origin,” is an
example of a constructive axiom and in the implicit form relies on the abstrac
tion of Potential Practicability because after each step it assumes the possibility
of constructing the next line segment bigger than the previous one. The number
Alexey Stakhov
THE MATHEMATICS OF HARMONY
370
of measurement steps in EudoxusArchimedes’ axiom, necessary for realization of
the condition nB >A, is always finite, but potentially with no limitation. Here we
can see the example of the constructive understanding of infinity as a potential
category. In this connection we can pay attention to the internal contradiction of
the classical mathematical measurement theory (and as a consequence, of the the
ory of real numbers), which assumes in the initial suppositions (the continuity
axioms) the coexistence of two opposite ideas about infinity, that is, the actual or
completed infinity in Cantor’s axiom and the potential or uncompleted infinity in
the EudoxusArchimedes’ axiom.
7.3. Evolution of the Infinity Concept
7.3.1. An Infinity Concept
The book Philosophy of Mathematics and Natural Science (1927, German)
by the famous Germany mathematician Hermann Weyl (18851955) begins with
the following words: “Mathematics is the science of infinity” [166]. Without a
doubt, the infinity concept permeates throughout all mathematics, because math
ematical objects are, as a rule, members of the classes or sets containing an un
countable set of elements of the same kind; examples include the set of natural
numbers, the set of real numbers, the set of triangles, etc. That is why the infin
ity concept is necessary under a strict analysis.
Starting a discussion of the infinity concept, we should say a little about
the concept of the “finite” that is opposite to the notion of “infinity.” Our
intellect determines that infinity is something that does not have an end, and
the finite is something that does have an end. At first sight, the concept of the
finite seems to us clear and selfobvious. However, in actuality it is not so
simple. If we consider the finite as a selfobvious concept, then infinity can be
seen as an unlimited accumulation of the finite. On the other hand, recognizing
the infinite division of the finite object (for example, the geometric line seg
ment that consists of points), we are compelled to state that the infinite is
made from the finite and the finite is made from the infinite.
The secret of the infinity in space and of the “immortality” in time has
always been the most highpowered catalyst that directs the human intellect
to the search for truth. David Hilbert, one of the most brilliant mathematicians
of the first half of the 20th century, said:
Chapter 7
Algorithmic Measurement Theory
371
“No other question has ever moved so profoundly the spirit of man; no other
idea has so fruitfully stimulated his intellect; yet no other concept stands in great
er need of clarification than that of the infinite” [quoted from E. Manor, To In
finity and Beyond, Boston, 1987].
7.3.2. The Origin of the Symbol ∞
The symbol for “infinity” in the form of the “horizontal 8” was introduced for
the first time in Arithmetic of Infinite Values by English mathematician John
Vallis in 1665. Some historians of mathematics give the following explanation
for the use of this infinity symbol: a similar symbol was used in Roman notation
and was designated as a thousand which was identified as “very much.” The Rus
sian historian of mathematics Gleiser in his book [150] gives, however, another
explanation of the origin of this symbol in Vallis’ book. According to his opin
ion, the symbol ∞ can be considered as two zeros connected among themselves.
This symbol was opposed by Vallis to the symbol “zero” (0) in the connection
with the following expressions given in Vallis’ book:
1 / 0 = ∞, 1 / ∞ = 0.
(7.6)
However, at the present moment, the expressions (7.6) are considered to be
an error. People should know and remember that the symbol ∞ does not mean
any number and does not have any numerical sense. Therefore, it is necessary to
use the symbols +∞ and −∞ with big caution; in particular, we cannot fulfill any
arithmetical operations similar to (7.6) with them.
7.3.3. Potential and Actual Infinity
Although according to Hermann Weyl, “infinity” is a fundamental concept of
mathematics, there is nevertheless no comprehensible definition of this important
concept in mathematics. Arithmetical and Geometric, Potential and Actual infini
ty are used in mathematics. Let us consider these concepts in greater detail.
A sequence of natural numbers
1,2,3,…
(7.7)
is the first and most important example of Arithmetical Infinity. Since Hegel,
the arithmetical infinity of natural numbers 1+1+1+ … i.e. the endless “itera
tion of the same” is called the “bad infinity.”
Geometric Infinity, for instance, consists of the unlimited bisection of a line
segment. Pascal wrote about geometric infinity as follows: “There is no geometer
who would not suppose, that any space can be divided ad infinitum. It is impossi
ble to be devoid of this, similar to a person, who cannot be without a soul. And
Alexey Stakhov
THE MATHEMATICS OF HARMONY
372
nevertheless there is no person who would understand that means an infinite di
visibility” [166]. Except for the distinction between arithmetical and geometric
infinity, there is a distinction between Actual and Potential infinity. For consid
eration of the difference between these concepts, once again, we address to the
sequence of natural numbers (7.7). We can suppose this sequence as if “complet
ed” sequence, that is, determined by all its members simultaneously. Such pre
sentation about “infinity” is named Actual Infinity.
However, the sequence of natural numbers (7.7) can also be considered as
the “developing” sequence that is generated according to the principle
n′=n+1.
(7.8)
This means that each natural number can be obtained from the previous
one by means of summation of 1. Such presentation of the “infinity” is named
Potential Infinity.
7.3.4. An Origin and Application of the Infinity Idea in the Ancient Greek
Mathematics
Mathematics was turned into a deductive science in Ancient Greece. Many
historians of mathematics believe that Greek mathematicians introduced into
mathematics for the first time the concept of Potential Infinity.
“Zeno’s paradoxes” played a major role in the development of the infinity
concept. They demonstrated so well the logical difficulty of the hypotheses about
the infinite division of geometric line segments and time intervals. Let us con
sider two of them, Dichotomy and Achilles and the Tortoise.
1. Dichotomy (“bisection” in Greek). In this antinomy Zeno asserts that
movement is impossible. Really, if a solid moves from point A to point B,
this solid must first traverse the line segments in 1/2, and before that 1/4, 1/
8, 1/16 ... of the distance between points A and B. However, the sequence of
such line segments is infinite. This means that one can never leave point A!
The paradox, which results in an insuperable logical impasse, is based on the
theorem that the sum of the infinite set of addends is finite.
2. Achilles and the Tortoise. Zeno asserts: “The fastmoving Achilles can never
catch up to the tortoise.” The proof comes to the following: let Achilles be n
times faster than the tortoise and let the initial distance between them be
equal to d. While Achilles overcomes this new distance, simultaneously with
him the tortoise can move forward on the distance d/n; while Achilles over
comes new distance, the tortoise can move forward on the new distance d/n2
and so on. Thus, the distance between Achilles and the tortoise will always be
more than 0, that is, Achilles never can catch up to the tortoise.
Chapter 7
Algorithmic Measurement Theory
373
Aristotle paid a great deal of attention to the concept of mathematical infin
ity. He strongly objected to the use of actual infinity in mathematics. The fol
lowing wellknown thesis is attributed to Aristotle: “Infinitum Actu Non Da
tur;” translated from Latin this means that the existence of actuallyinfinite ob
jects in mathematics is impossible.
7.3.5. Cantor’s Theory of Infinite Sets
The German mathematician George Cantor (18451918) is considered to be
the prime disrupter of tranquility in 19th century mathematics. The history of
set theory is quite different from the history of other areas of mathematics. For
most areas, we can outline a long process, in which the ideas are evolving until
the ultimate flash of inspiration produces a discovery of major importance, often
in many mathematicians somewhat synchronously. That nonEuclidean geome
try was discovered by Lobachevski and Bolyai independently at the same time
is a brilliant example of such synchronous inspiration. As for set theory, it was
the creation of one person, George Cantor.
Cantor’s main idea is the study of infinite sets as actualinfinite sets. The
idea of onetoone correspondence between the elements of the comparable
sets was used by Cantor in his research of infinite sets. If we can establish such
correspondence between the elements of two sets, we can say that the sets have
the same Cardinality, that is, they are equivalent. Cantor wrote “In the case of
the finite sets the cardinality coincides with the number of the elements.” That
is why, the cardinality is also named the cardinal (quantitative) number of the
given set. This approach resulted in many paradoxical conclusions being in sharp
contradiction to one’s intuition. So, in contrast to the finite sets that comply
with the Euclidean axiom, “The whole is more than its parts,” the infinite sets do
not comply with this axiom. For example, it is easy to prove that the set of
natural numbers and some of its subsets are equivalent. In particular, we can
establish the following onetoone correspondence between the sets of natural
numbers and the even numbers:
1 2 3 ... n ...
7 7 7
7
.
2 4 6 ... 2n ...
This feature of the infinite sets can be used for their definition: a set is named
infinite if the set is equivalent in extant to one of its subsets. A set is finite if the set
is not equivalent to any of its subsets. Any set equivalent to the set of natural
numbers is named Denumerable because all of its elements can be numbered.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
374
The most amazing discovery was made by Cantor in 1873. He proved that
all three of the most characteristic sets (natural numbers, rational numbers, and
algebraic numbers) have one and the same cardinality, that is, the sets of ratio
nal and algebraic numbers are denumerable. Cantor also proved that the cardi
nality of real numbers is greater than the cardinality of natural numbers.
From the above consideration it is possible to conclude that in contrast to the
majority of his predecessors, Cantor had undertaken for the first time a deep re
search into mathematical infinity that resulted in new and unexpected outcomes.
7.3.6. Antinomies of Cantor’s Theory of Infinite Sets
It seemed for many mathematicians, that Cantor’s theory of infinite sets pro
duced a revolution in mathematics. The end of the 19th century culminated in
the recognition of Cantor’s theory of infinite sets. French mathematician Jacques
Hadamard in his 1897 speech at the First International Congress of Mathemati
cians in Zurich officially proclaimed the settheoretical ideas as the basis of math
ematics. In his lecture, Hadamard emphasized that the most attractive reason of
Cantor’s set theory consists of supporting the fact that for the first time in math
ematics history the classification of sets was made on the basis of a new concept
of “cardinality” and the amazing mathematical outcomes inspired mathemati
cians to new and surprising discoveries.
However, very soon the “mathematical paradise” based on Cantor’s set
theory was destroyed. In the first few years after the First International
Congress of Mathematicians, the paradoxes or antinomies appeared in set
theory. Cantor discovered the first paradoxes at the end of the 19th century.
Other paradoxes were discovered later by other scientists. These paradoxes be
came the basis for the next, or third (after the discovery of incommensurable
line segments and substantiation of the limit theory) crisis in the foundation of
mathematics. One of them, discovered by the English philosopher and mathe
matician Bertrand Russell in 1902, was connected with the foundations of set
theory and the concept of the set of all sets. Each of the usual sets, which we met
until now, does not contain itself as an element; so, for example, the set of all
natural numbers is not a natural number. We name such sets Ordinary Sets.
However, there are also such exotic sets that contain themselves as their own
elements, for example, the set of all sets. We name such sets Nonordinary Sets.
Consider the set S of all ordinary sets and ask the following question: is S an
element of S? Alternatively, is the set S ordinary or nonordinary set? Suppose
that S is an element of S; then it is necessary to recognize that S is a nonordinary
set. However, as S contains only ordinary sets, this means that S should also be
Chapter 7
Algorithmic Measurement Theory
375
an ordinary set. Thus, from this supposition it follows that S is an ordinary set,
that is, we have a logical contradiction. Now, suppose that S is not an element of
S, that is, S is the ordinary set; then S should be in S, as all ordinary sets. Thus,
from the supposition that S is the ordinary set it follows that S is the nonordi
nary set; again, we have contradiction!
Russell demonstrated this contradiction, known as the Barber Paradox, with
the example of the village barber, who gave a promise to shave those and only
those village inhabitants, who do not shave themselves. We can ask the ques
tion: Does the barber shave himself? If he shaves himself, this means that there
by he includes himself into the set of inhabitants, who shave themselves, and
therefore he should not shave himself; if he does not shave himself, this means
that he belongs already to those, who do not shave themselves, that is, he should
shave himself. We again have a logical contradiction which of course is unac
ceptable in mathematics!
7.3.7. Is Cantor’s Actual Infinity the Greatest Mathematical
Mystification?
A discovery of the antinomies in Cantor’s set theory caused a new crisis in
the foundations of mathematics. Various attempts were made to overcome
this crisis including Constructive Analysis in mathematics the most radical of
them. The representatives of constructive analysis saw the main reason for
paradoxes in Cantor’s theory of sets to be the use of the concept of “actual
infinity.” Russian mathematician Markov, one of the brightest representa
tives of constructive analysis, wrote: “It is impossible to imagine an endless
process as a completed process without rough violence to our intellect, in the
rejection of such inconsistent fantasies.” [167]
On the other hand, probably the works of Russian mathematician Alex
ander Zenkin took the final step in the dispute over Cantor’s set theory. Zenkin
found logical contradictions and errors in Cantor’s set theory. The essence of
Zenkin’s approach, presented in a few main publications [168], consists of the
fact that for the first time he formulated an almost obvious fact – the concept
of potential infinity (PI) in the Aristotelian form – “all infinite sets are poten
tial sets,” and the concept of actual infinity (AI) in Cantor’s form – “all infinite
sets are actual sets” are AXIOMS, that is, they cannot be proved or refuted,
but they can be either accepted, or rejected. For this reason, any discussions
about “existence” or “nonexistence” of the “actual infinity,” which goes back
to Aristotle, cannot convince anyone of anything. Here we can provide an
analogy with the history of the 5th Euclidean axiom about parallel lines. Howev
Alexey Stakhov
THE MATHEMATICS OF HARMONY
376
er, there is also an essential difference – the correctness of the nonEuclidean ge
ometry was proven – however, the correctness of Cantor’s theory of sets
based on “actual infinity” is not proven. Moreover, the antinomies of Cantor’s
theory of sets demonstrate that this theory is self contradictory.
After a thorough analysis of Cantor’s continuum theorem, in which Zen
kin gave the logical substantiation for legitimacy of the use of the “actual
infinity” in mathematics, he derived the following unusual conclusion:
1. Cantor’s proof of this theorem is neither a mathematical proof in Hilbert’s
sense nor in the sense of classical mathematics.
2. Cantor’s conclusion about nondenumerability of the continuum is a jump
through a potentially infinite stage, that is, Cantor’s reasoning contains the
fatal logical error of “unproved basis” (a leap to a “wishful conclusion”).
3. Cantor’s theorem does, in fact, strictly mathematically prove the potential,
that is, the unfinished character of the infinity of the set of all “real numbers.”
Thus, Cantor proves mathematically the fundamental principle of classical
logic and mathematics: “Infinitum Actu Non Datur” (Aristotle).
According to the opinion of Alexander Zenkin [168], Aristotle’s famous the
sis “Infinitum Actu Non Datur” – that is, the assertion about the impossibility
of the existence of actuallyinfinite objects – had been supported during the last
2300 years by Aristotle’s great followers Leibniz, Cauchy, Gauss, Kronecker,
Poincare, Brouwer, Vail, Lusin and by many other famous founders of classical
logic and modern classical mathematics! Every one of them professionally stud
ied the problem of mathematical infinity, and there were not be any doubts, that
they understood the true nature of the infinity in general no worse than Cantor
did. Especially, if we take into consideration the important fact, that infinity
does not depend on progress of technological equipment of science, infinity nev
er was and never becomes an object of the technological research, because no
contemporary computers, never, by definition, would ever be able to finish an
enumeration of all elements of natural series 1, 2, 3, … .
For this reason – Alexander Zenkin asserted – all discussions about the pos
sibility or impossibility of actual infinity during two millennia down to
Cantor had only a speculative character. The infinity does not depend on the
progress of science and technology!
Alexander Zenkin wrote [168]: “Why was such an analysis of Cantor’s
theorem not executed in time, i.e. at the end of the 19th century? It is a very
crucial topic for fundamental research in the field of the psychology of scien
tific knowledge.”
Thus, there is the impression that the long history (starting from Greek
science) of the study of the infinity concept as one of the fundamental mathe
Chapter 7
Algorithmic Measurement Theory
377
matical concepts is approaching its completion. The antinomies of Cantor’s set
theory, which caused the contemporary crisis in the foundations of mathematics,
showed that the concept of actual infinity could not be a reliable basis for mathe
matical reasoning, because from the point of view of the representatives of con
structive analysis the “actual infinity” concept is internally selfcontradictory. On
the other hand, the research of Russian mathematician Alexander Zenkin [168]
testifies the existence of logical errors in Cantor’s theorems, and this fact gives one
the right to doubt the accuracy of Cantor set theory.
Thus, the concept of potential infinity introduced in the Greek mathe
matics can be the only real base for mathematical reasoning. In this connec
tion Aristotle’s thesis “Infinitum Actu Non Datur” should become the main
axiom for the creation of new, constructive mathematics.
7.4. A Constructive Approach to Measurement Theory
7.4.1. The Rejection of Cantor’s Axiom from Mathematical
Measurement Theory
As the concept of “actual infinity” is an internally contradictory notion (“the
completed infinity”), this concept cannot be a reasonable basis for the creation
of constructive mathematical measurement theory. If we reject Cantor’s axiom,
we can try to construct mathematical measurement theory on the basis of the
idea of potential infinity, which underlies the EudoxusArchimedes’ axiom. By
referring to the measurement theory, this means, that the number of steps for
the given measurement is always finite, but potentially unlimited. The accep
tance of the given approach at once results in the occurrence of the fundamental
ly irremovable measurement error called Quantization Error.
The constructive approach to measurement theory results in a change of
the purpose of measurement theory. One of the essential moments at the proof
of the equality (7.5) is a choice of the Measurement Algorithm, by means of
which we carry out the formation of “nested segments” from the measurement
unit V. With the actually infinite time, the measurement algorithm does not
influence the measurement result q and its choice is arbitrary. With the given
time of measurement – that is defined by the number of measurement steps n
– and with the other equal conditions, the distinction between the measure
ment algorithms with respect to the “reached” measurement exactness appears.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
378
With these conditions, the second constructive idea about the “efficiency” of
the measurement algorithms comes into effect and the Problem of the Synthe
sis of the Optimal Measurement Algorithms is put forward as a central problem
of the Constructive (Algorithmic) Measurement Theory.
7.4.2. BashetMendeleev’s Problem
The constructive approach, at once, results in the optimization problem in
measurement theory which was not a topical problem for classical measurement
theory. By studying the optimization problem in the measurement theory, we
come unexpectedly to a combinatorial problem known as a Problem about the
Choice of the Best Weights System. For the first time, this prob
lem appeared in the 1202 in Fibonacci’s Liber Abaci. From Fi
bonacci’s work this problem moved to the 1494 book Summa
de Arithmetica, Geomeytria, Proprtioni et Proportionalita by
Luca Pacioli. After Pacioli’s work, this problem again appeared
in the 1612 book Collection of the Pleasant and Entertaining
Problems by the French mathematician Claude Gaspar Bachet
Claude Gaspar
de Meziriac.
Bachet de Meziriac
Claude Gaspar Bachet was born in BourgenBresse in Sa
(15811638)
voy (France). This was a region, which in different periods belonged to France,
Spain or Italy. Therefore, the life of Claude Gaspar Bachet is connected with
France, Spain and Italy of that period. He was a writer of books on mathemati
cal puzzles and tricks that became a basis for all the books on mathematical puz
zles. Bachet’s 1612 book Collection of the Pleasant and Entertaining Problems
contains different numerical problems. Fibonacci’s “weighing” problem is one of
them. This problem was formulated as follows: “What is the least number of
weights used to weigh any whole number of pounds from 1 to 40 inclusively by
using a balance, if the weights can be placed on both balance cups?”
In the Russian historicalmathematical literature
[169] Fibonacci’s “weighing” problem is known also un
der the name of the BachetMendeleev problem in honor
of Bachet de Meziriac and the famous Russian chemist
Dmitri Mendeleev. However, this raises the question:
why did the great Russian chemist become interested in
the “weighing” problem? The answer to this question re
quires some littleknown facts from the life of the great
scientist. In 1892, Mendeleev was appointed the director
Dmitri Mendeleev
of the Russian Depot of Standard Weights, which, accord
(18341907)
Chapter 7
Algorithmic Measurement Theory
379
ing to Mendeleev’s initiative, was transformed in 1893 to the Main Board of
Weights and Measures of Russia. Mendeleev remained its director until the end
of his life. Thus, the final stage of Mendeleev’s life (since 1892 until his death in
1907) was connected with the development of metrology. During this period,
Mendeleev was actively involved in various problems connected with measures,
measurement and metrology; the “weighing” problem was one of them. Men
deleev’s contribution to the development of metrology in Russia was so great,
that he was named “the father of Russian metrology” and the “weighing” prob
lem was named the BachetMendeleev problem.
The essence of the problem consists of the following [169]. Suppose, we need
to weigh on a balance any integervalued weight Q in the range from 0 up to Qmax
by using the n standard weights {q1, q2, …, qn}, where q1=1 is a measurement unit;
qi=ki×q1; ki is any natural number. It is clear that the maximal weight Qmax is
equal to the sum of all the standard weights, that is,
Qmax = q1+ q2+ …+ qn = (k1+ k2+…+ kn) q1.
(7.9)
We have to find the Optimal System of Standard Weights, that is, such
standard weight system, which gives the maximal value of the sum (7.9) with
a given measurement unit q1=1 among all possible variants.
There are two variants of the solution to the BachetMendeleev problem. In
the first case, the standard weights can be placed only on the free cup of the
balance; in the second case, they can be placed on both cups of the balance. The
Binary Measurement Algorithm with the binary system of standard weights {1,
2, 4, 8,…, 2n1} is the optimal solution for the first case. It is clear that the binary
measurement algorithm generates the binary method of number representation:
n −1
Q = ∑ ai 2 i ,
i =0
(7.10)
where ai∈{0,1} is a binary numeral.
Note that the binary numerals 0 and 1 have a precise measurement interpre
tation. The binary numerals 0 and 1 encodes one of the two possible positions of
the balance; thus, ai=1, if the balance remains in the initial position after putting
a new standard weight on the free cup, otherwise ai=0.
The ternary system of the standard weights {1, 3, 9, 27,…, 3n1} is the optimal
solution to the second variant of the “weighing” problem when we can put the
standard weights on both cups of the balance. It is clear that the ternary system
of the standard weights results in the ternary method of number representation:
n −1
Q = ∑ bi 3i ,
i =0
where bi is a ternary numeral that takes the values {1, 0, 1}.
(7.11)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
380
Hence, by solving the “weighing” problem, Fibonacci and his followers found a
deep connection between measurement algorithms and positional methods of num
ber representation. This is the key idea of algorithmic measurement theory [20,
21], which goes back to the “weighing” problem for its origin. Thus, from the measure
ment problem we come unexpectedly to the positional method of number representa
tion the greatest mathematical discovery of Babylonian mathematics.
7.4.3. Asymmetry Principle of Measurement
The principles of Finiteness and Potential Feasibility of measurement, under
lying the constructive measurement theory, are “external” with respect to the
measurement. They have such a general character that there is a danger of their
reduction to some trivial result (for example, to the above binary measurement
algorithm that leads the measurement to the consecutive comparison of the measur
able weight with the binary standard weights 2n1, 2n2, …, 20).
For obtaining nontrivial results, the methodological basis of the con
structive (algorithmic) theory of measurement should be added to by a cer
tain general principle, which follows from the essence of measurement. Such
principle follows directly from the analysis of the binary measurement algo
rithm, which is the optimal solution to BachetMendeleev’s problem.
+
a)
...
n1
2
Q
n1
2 <Q
2
n1
2
n-2
... 2
0
b)
...
n1
n1
2
2 tQ
2
n2
... 2
0
2n1
+
...
c)
n2
2
n1
2
Figure 7.3. Asymmetry Principle of Measurement
2
n2
... 2
0
Chapter 7
Algorithmic Measurement Theory
381
We will analyze the above binary measurement algorithm by means of the use
of the balance model (Fig. 7.3). This analysis allows for finding a measurement
property of general character for any thinkable measurement, based on the com
parison of the measurable weight Q with the standard weights.
We shall examine the weighing process of the weight Q on the balance, by
using the binary standard weights. On the first step of the binary algorithm the
largest standard weight 2n1 is placed on the free cup of the balance (Fig. 6.3a).
The balance compares the unknown weight Q with the largest standard weight
2n1. After the comparison, we can get two situations: 2n1<Q (Fig. 7.3a) and
2n1≥Q (Fig. 6.3b). In the first case (Fig. 7.3a), the second step is to place the
next standard weight 2n2 on the free cup of the balance. In the second case
(Fig.7.3b), the weigher should perform two operations. First, one should
remove the previous standard weight 2n1 from the free cup of the balance
(Fig. 7.3b), after that, the balance should return to the initial position (Fig.
7.3c). After returning the balance to the initial position, the next standard
weight 2n2 should be placed on the free cup of the balance (Fig. 7.3c).
As we can see, both cases differ in their complexity. In the first case, the weigher
fulfils only one operation, that is, he adds the next standard weight 2n2 on the free
cup of the balance. In the second case, the weigher’s actions are determined by
two factors. First, he has to remove the previous standard weight 2n1 from the free
cup of the balance, and after that, the balance is back to the initial position. After
that, the weigher has to place the next standard weight on the free cup of the bal
ance. Thus, for the first case the weigher has to perform only one operation. How
ever, for the second case the weigher has to perform two sequential operations:
(1) remove the previous standard weight 2n1 from the free cup;
(2) place a new standard weight 2n2 on the free cup of the balance.
In the second case, the weigher’s actions are more complicated in comparison
with the first case because the weigher has to remove the previous standard weight
taking into consideration the time necessary for the balance to return to its initial
position. By synthesizing the optimal measurement algorithms, we have to consider
these new data, which influence the process of measurement. We name this discov
ered property of measurement, the Asymmetry Principle of Measurement [20].
7.4.4. A New Formulation of the BachetMendeleev Problem
Now, let us introduce the above measurement property into the Ba
chetMendeleev problem. For this purpose we will examine a measurement
as a process, running during discrete periods of time; let the operation to
“add the standard weight” be carried out within one unit of discrete time
Alexey Stakhov
THE MATHEMATICS OF HARMONY
382
and the operation to “remove the standard weight” (which is followed by re
turning the balance into the initial position) be carried out within p units of
discrete time, with p∈{0,1,2,3,…}.
It is clear that the parameter p simulates the Inertness of the balance. Note that
the case p=0 corresponds to the “ideal situation” when we neglect the inertness of the
balance. This case corresponds to the classical BachetMendeleev problem. For the
other cases p >0, we have some new variants of the BachetMendeleev problem, which
are studied in the algorithmic measurement theory [20, 21]. Below we study other
generalizations of the BachetMendeleev problem.
7.5. Mathematical Model of Measurement
7.5.1. A Notion of the “Indicator Element”
If we put forward a problem to create the Algorithmic Measurement The
ory, we should define more exactly, what is a Measurement, what is the Purpose
of Measurement, what is a Measurement Algorithm, and How are the Measure
ment Algorithms carried out.
First of all, we should note that, if we wish to measure something, we should
know the range of the measurable magnitude. When we discuss “mathematical
measurement,” we are distracted from the physical nature of measurable magni
tude. We will present a Range of Measurable Magnitude in the form of the geo
metric line segment AB. Clearly the measurable magnitude is one of the possible
sizes of the magnitude that belong to the range of measurement, that is, prior to
the measurement, a certain “indeterminacy” about the measurable size exists be
cause otherwise the measurement would be senseless. We will represent this situ
ation of “indeterminacy” by the “unknown” point X, which belongs to the line
segment AB.
Now, let us formulate a purpose for the measurement. The purpose of the
measurement is to find the length of the “unknown” segment AX. In practice
this purpose is carried out by means of special devices, for example, “balances”
or “comparators.” The “comparators” perform a comparison of the measurable
magnitude with some “measures,” generated from the “measure unit.” In de
pendence on the result of comparison, the device “indicates” the result of com
parison in the form of a binary signal 0 or 1. Thus, the essence of the measure
ment consists of Consecutive Comparisons of the measurable magnitude with
some “measures,” which are formed at each step of the measurement. More
Chapter 7
Algorithmic Measurement Theory
383
over, each step of the measurement depends on the results of comparison on the
preceding steps.
In order to model the process of comparison of the measurable segment
AX with the “measures,” we introduce the important concept of the Indicator
Element (IE), which is an original model of the “comparator” or “balance,” the
basic tool of any measurement. Suppose that every IE can be put to any
“known” point C of the segment AB. The indicator element provides the
information about the mutual positioning the “unknown” point X and the
“known” point C. If the IE is to the right of the point X, it “generates” the
binary signal 0; otherwise, the binary signal 1.
7.5.2. The (n, k, S)algorithms
Some Conditions or Restrictions S (that follow, for example, from the “Princi
ple of Asymmetry of Measurement”) can be imposed on the measurement pro
cess; thus, by means of the restrictions S the “inertness” property of the balance
can be taken into consideration.
A
C
C
C X C
C
B
By using a concept of the in
...
...
dicator elements, we can describe
Figure 7.4. A geometric model of measurement
the process of measurement in
terms of geometry. Suppose we
have a line segment AB with some “unknown” point X (Fig. 7.4).
The problem is to find the length of line segment AX. This is performed by
means of the k “Indicator Elements” (IE). After we enclose the jth IE (j=1,2,3,…,k)
at some point Cj, the line segments AX and ACj are compared, i.e. the relations “less
than” (AX< ACj) or “greater than or equal” (AX≥ACj) are defined. The relations
“less than” and “greater than or equal” are encoded by the binary numerals 0 and 1,
which are generated by the IE in the process of measurement. The problem of the
measurement of a line segment AX by the k IE’s is reduced to decreasing the Inde
terminacy Interval about X according to the IE“indications.”
A measurement procedure consists of the Measurement Steps: in the first
step the k IE’s are enclosed to some points of the line segment AB, and the
indeterminacy interval about X is diminished to the line segment A1B1 ac
cording to the IE“indications”; in the next step the IE’s are enclosed to the
points of the line segment A1B1, and so on. The restrictions S are imposed on
the measurement procedure. For the given restrictions S and the given n and
k, the system of formal rules, which strictly define the points of the k IE’s
enclosures for each step in dependence on the IE“indications” on the preced
ing steps, is called the (n,k,S)algorithm.
1
2
j
j+1
k
Alexey Stakhov
THE MATHEMATICS OF HARMONY
384
7.5.3. The Optimal (n, k, S)algorithms
A model of the indicator elements reduces the measurement problem to the
problem of searching the point X (its coordinate is equal to the length of the line
segment AX) within the line segment AB by means of the use of the k IE’s in n
steps. The above approach allows one to use the methods of onedimensional
search for the synthesis of the optimal measurement algorithms [20].
Let us consider the (n,k,S)algorithm, which acts on the line segment AB.
Some “indeterminacy interval” ∆i, which contains the point X on the last step
of the algorithm, is the result of the (n,k,S)algorithm for the certain point X.
Let us consider the action of the algorithm for all possible points X∈AB. As
a result we obtain the set of line segments {∆i} called the Partition P of the Line
Segment AB, i.e.
P={∆1, ∆2 , …, ∆i , …, ∆N }.
(7.12)
For the general case, the partition (7.12) consists of the N line segments
of the kind ∆i. They are not equal to one another in the general case. Besides,
there is the following correlation for the partition P:
AB=∆1+∆2+…+∆N.
(7.13)
We choose from the partition (7.13) the largest line segment ∆max and
define the effectiveness of the (n,k,S)algorithm by means of the line segment
∆ max . Consider the ratio
AB
T=
,
(7.14)
∆ max
which is called the (n,k)exactness of the (n,k,S)algorithm.
In accordance with (7.14), each (n,k,S)algorithm is characterized by the
number T, which is some numerical evaluation of the effectiveness of the (n,k,S)
algorithm. The availability of such a number T allows one to compare the dif
ferent (n,k,S)algorithms according to their effectiveness. By using the defini
tion (7.14), we can introduce the notion of an Optimal (n,k,S)algorithm.
Definition 7.1. For some given n, k, and S, the (n,k,S)algorithm is called
optimal, if it provides a maximal value of the (n,k)exactness T given by (7.14)
among all possible variants.
Note that this definition is based on the socalled Minimax Principle [20].
This principle is used widely in the modern theory of optimal systems. Ac
cording to this principle we refer some strategy to the “optimal” strategy, if it
provides the maximal value of the efficiency criterion for the worst case.
It is easy to prove [20] that the optimal (n,k,S)algorithm divides the line
segment AB into N equal parts, i.e.
Chapter 7
Algorithmic Measurement Theory
385
∆1 = ∆ 2 = ... = ∆ N = ∆.
(7.15)
For the case (7.15) we have the following expression for the (n,k)exactness:
AB
T=
= N.
(7.16)
∆
Thus, in accordance with (7.16) the ( n, k ) exactness of the optimal (n,k,S)
algorithm coincides with the number N of the quantization levels that are pro
vided by the optimal (n,k,S)algorithm.
7.6. Classical Measurement Algorithms
Over the millennia, the practice of human measurement found a number of
measurement algorithms that are widely used today in mathematics and mea
surement technology. The most common among them include the following:
Binary Algorithm, Counting Algorithm, Ruler Algorithm. We will study these
algorithms by using the above “indicator” model of measurement (Fig. 7.4).
7.6.1. The “Binary” Algorithm
The essence of the “Binary” Algorithm is demonstrated in Fig. 7.5. It con
sists of 3 steps (n steps in the general case) and uses only IE (k=1). By using 3
steps, the algorithm divides the initial line segment [0,8] into 8 equal parts.
The First Step is to enclose the IE to the middle of the initial line segment
[0,8], i.e. to point 4. After the first step there appear two situations in depen
dence on the IE“indication,” [0,4] and [4,8].
The Second Step:
(a) If the IE“indication” in point 4 is equal to 0 (the IE appears to the left),
this means that the “unknown” point X is at the line segment [0,4]. For this
situation the second step of the “binary” algorithm is to enclose the IE to the
middle of the line segment [0,4], that is, to point 2.
(b) If the IE“indication” in point 4 is equal to 1 (the IE appears to the right),
this means that the “unknown” point X is at the line segment [4,8]. For this
situation the second step of the “binary” algorithm is to enclose the IE to the
middle of the line segment [4,8], i.e. to point 6.
It is clear that after the second step four situations may appear as
shown in Fig. 7.5.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
386
8
4
0
1st step
0
2
4
4
6
8
2d step
0
1
2
2
3
4
4
5
6
3d step
6
7
8
Figure 7.5. The “binary” algorithm
The Third Step. We can see from Fig. 7.5 that after the second step of the
“binary” algorithm we have the four “indeterminacy intervals” [0,2], [2,4], [4,6],
[6,8]. The third step of the “binary” algorithm is to enclose the IE at the middle
of one of the “indeterminacy intervals” [0,2], [2,4], [4,6], [6,8]. It follows from
Fig. 7.5 that the “binary” algorithm provides the following partition of the initial
line segments [0,8]:
P={[0,1], [1,2], [2,3], [3,4], [4,5], [5,6], [6,7], [7,8]},
that is, it divides the line segment [0,8] into 8 equal parts.
It is clear that the above “binary” algorithm is the (3,1,S)algorithm and its
(3,1)exactness is equal to T=23=8. It is easy to prove that the nstep “binary”
algorithm provides the (n,1)exactness given by the following formula:
T = 2n .
(7.17)
7.6.2. The “Counting” Algorithm
Now, let us consider the socalled Counting Algorithm widely used in mea
surement practice. The “counting” algorithm (see Fig. 7.6) consists of 3 steps (n
steps in the general case) and uses only the IE (k=1). It divides the initial line
segment [0,4] into 4 equal parts in 3 steps.
The First Step is to enclose the IE at the point 1. After the first step there appear
two situations, [0,1] and [1,4], depend on the IE“indication” in point 1.
Chapter 7
Algorithmic Measurement Theory
387
The Second Step:
(a) If the IE“indication” in
0
1
point 1 is equal to 0 (the IE appears
4 to the left), then the “unknown”
1
2
2d step
point X is at the line segment [0,1].
1
2
For this situation the measurement
process is over because the Xcoor
3
4
2
3d step
dinate (X∈[0,1]) is defined with
“exactness” equal to measurement
Figure 7.6. The “counting” algorithm
unit 1.
(b) If the IE“indication” in point 1 is equal to 1 (the IE appears to the
right), this means that the “unknown” point X is at the line segment [1,4].
For this situation the measurement process will continue and the IE is en
closed at point 2.
The Third Step. We can see from Fig. 7.6 that on the third step the IE is
enclosed at point 3, which is from point 2 on the distance equal to the measure
ment unit 1.
From the example in Fig. 7.6 we can find the following general rule for the
“counting” algorithm. If on the ith step of the “counting” algorithm, which
acts at the line segment AB, the IE was enclosed at point C, then we have two
situations on the (i+1)th step. If the IE appears at point C to the left, then the
measurement procedure is over. If the IE appears at point C to the right, then
we get the “indeterminacy interval” CB and the IE on the (i+1)th step is en
closed at point C′=C+1.
It is clear that the above “counting” algorithm is the (3,1,S)algorithm
and its (3,1)exactness is equal to T=4. It is easy to prove that the nstep
“counting” algorithm has the (n,1)exactness given by the following formula:
1st step
0
1
T = n +1.
4
(7.18)
7.6.3. The “Ruler” Algorithm
Let’s consider one more example of the measurement algorithm widely
used in measurement practice. The latter underlies the traditional measure
ment ruler. We call this algorithm the “Ruler” Algorithm. The essence of the
latter is demonstrated in Fig. 7.7.
The “ruler” algorithm is performed in 0
1
3
2
4
1 step and uses the k IE (the 3 IE in Fig.
7.7). The algorithm divides the initial
Figure 7.7. The “ruler” algorithm
line segment into
Alexey Stakhov
THE MATHEMATICS OF HARMONY
388
(7.19)
T = k +1
equal parts (into 4 parts in Fig. 7.7).
The formula (7.19) gives the (1, k)exactness of the “ruler” algorithm. We
can see that the latter refers to a class of the (1,k,S)algorithms.
7.6.4. The Restriction S
The above definition of the (n,k,S)algorithm includes a concept of the
Restriction S. Clearly, any “restriction” limits the efficiency of the algorithm.
However, each “restriction” (as show below) can result in rather unusual
measurement algorithms, which are of theoretical and practical interest.
We start with the elementary “restrictions,” which can be found in the
comparative analysis of the classical measurement algorithms, in particular,
the “binary” algorithm and the “counting” algorithm. Compare two 3step
(n,k,S)algorithms, shown in Figs. 7.5 and 7.6. Both algorithms consist of 3
steps and use only the IE (k=1). What is the difference between them? The
difference consists in the character of the movement of the IE along the line
segment AB. Consider the restriction S for the “binary” algorithm in Fig. 6.6.
Remember that the restriction S is imposed upon the movement of the IE along
the line segment AB. We can see that the IE for the “binary” algorithms may
be enclosed at each step to the left or to the right of the “unknown” point X.
One may say that the movement of the IE for this case is performed with
out any “restrictions.” Denote the restriction S for the “binary” algorithm
by S≡0. We will call the class of (n,k,S)algorithms, which each individually
satisfy the restriction S≡0, the (n,k,0)algorithms.
Consider the restriction S for the “counting” algorithm. We can see that
for this case the IE is moving along the line segment [0,4] in only one direc
tion, namely, from the point 0 to the point 4. We denote the restriction of this
kind by S≡1 and call the class of (n,k,S)algorithms satisfying the restriction
S≡1, the (n,k,1)algorithms.
Note that the restrictions S≡0 and S≡1 are not the only possible “re
strictions.” Below we will study the “restriction” S, which follows from the
above formulated Asymmetry Principle of Measurement. It is important to
emphasize, that each “restriction” results in the development of one or an
other class of the new measurement algorithms, which can be of theoretical
or practical interest; therefore, a search of reasonable “restrictions” impos
able on the measurement algorithm, is an important problem in algorithmic
measurement theory.
Chapter 7
Algorithmic Measurement Theory
389
7.6.5. Connection between Measurement Algorithms and Positional
Number Systems
There is a deep connection between measurement algorithms and positional
methods of number representation. In fact, the “binary” algorithm generates
the binary representation of numbers:
(7.20)
A = an −1 2n −1 + an −2 2n −2 + ... + ai 2i + ... + a1 21 + a0 20 ,
where ai is a binary numeral {0,1}; 2i is the weight of the ith digit (i=0,1,2, …, n1).
The formula (7.20) has the following “measurement” interpretation. The
binary numerals an1, an1, …, a0 are the IE“indications” on the first, second, ... ,
nth step of the “binary” algorithm, respectively.
One can readily see that the “counting” algorithm “generates” the
following method of number representation that underlies the Euclidean
definition of natural number:
N = 1
+ 1 + ... + 1.
N
(7.21)
The latter is well known as a Unitary Code of the number N.
7.7. The Optimal Measurement Algorithms Originating Classical Position
al Number Systems
7.7.1. A General Method for the Synthesis of the Optimal (n,k,S)
algorithms
For the synthesis of the optimal (n,k,S)algorithm we use the Method of
Recurrence Relations [170]. To obtain the recursive relation for (n,k)exact
ness T of the optimal (n,k,S)algorithm, we first formulate an Inductive As
sumption: for any arbitrary n, k and S there is the optimal (n,k,S)algorithm,
which divides the line segment AB into T equal line segments of length D, where
T is the (n,k)exactness of the optimal (n,k,S)algorithm. For a given restriction
S, the (n,k)exactness T depends on n and k. We can present this dependence in
the form of some function of n and k, i.e.
T=F(n,k).
It is clear if the length D=1, then the line segment
(7.22)
AB=T=F(n,k).
(7.23)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
390
Consider the first step of the optimal (n,k,S)algorithm, which acts on the
line segment AB (Fig. 7.8).
A
C1
C2
Cj
...
Cj+1
Ck
B
...
Figure 7.8. The first step of the optimal (n, k, S)algorithm
Let the first step be to enclose the k IE’s at some points C1,C2,…,Cj,Cj+1, …, Ck.
Then, according to the IE“indications” the following (k+1) different situations
may result:
(1) X ∈ AC1 ; ( 2 ) X ∈C1C2 ; ...; ( j + 1) X ∈C jC j +1 ; ...; ( k + 1) X ∈Ck B.
(7.24)
The second step of the optimal (n,k,S)algorithm is to enclose the IE’s at the
points of the new “indeterminacy interval,” which is one of the line segments
(7.24). It is clear that for this situation we can use the optimal (n1)algorithm.
However, the number of the IE’s on the second step depends on the restriction
S. Suppose, that in accordance with the restriction S we can use only the r IE’s
(r≤k) on the second step of the algorithm. Note that the number r may vary for
the different situations (7.24).
Suppose that the line segment CjCj+1 is the “indeterminacy interval” on the
second step. Since on the second step we have r IE’s and (n1) steps of the algo
rithm, we can use the optimal (n1,r,S)algorithm on the line segment CjCj+1. In
accordance with the inductive assumption, the (n1,r,S)algorithm will divide
the line segment CjCj+1 into F(n1,r) equal line segments of length ∆=1, i.e.
CjCj+1 = F(n1,r).
Using the equality (7.13), we can write:
(7.25)
AB=AC1+ C1C2+…+ CjCj+1+…+CkB.
(7.26)
Using (7.23) and (7.25), we can get the recursive relation for the calcula
tion of the function (7.23).
7.7.2. The Optimal (n,k,0)algorithms
Now, we can use the above method for the synthesis of the optimal (n,k,0)
algorithm. Recall that the restriction S≡0 means what each IE may be en
closed at any arbitrary point of the line segment of AB on each step.
According to the above method, one of the situations (7.24) appears after the
first step of the optimal (n,k,0)algorithm. Consider the situation: X∈CjCj+1, which
can appear after the first step of the algorithm. Then in accordance with the restric
tion S≡0, the second step of the optimal (n,k,0)algorithm is to enclose all k IE’s at
the points of the new line segment CjCj+1. If we use the (n1,k,0)algorithm on the
Chapter 7
Algorithmic Measurement Theory
391
“indeterminacy interval” CjCj+1, then in accordance with the inductive assumption we
can divide the line segment CjCj+1 into F(n1,k) equal parts ∆=1, i.e.
(7.27)
CjCj+1 = F(n1,k).
The expression (7.27) is valid for any arbitrary line segment from the set
(7.24). Then, by using (7.26), we obtain the following recursive relation for
the calculation of the function F(n,k):
F(n,k)=(k+1)F(n1,k).
(7.28)
The recursive function (7.28) can be expressed in explicit form. For this
purpose, we can use the recursive relation
F(n1,k)=(k+1)F(n2,k)
for the representation of (7.28) in the following form:
F(n,k)=(k+1)(k+1)F(n2,k).
By continuing this process, we can represent the function (7.28) in the fol
lowing form:
F(n,k)=(k+1)n1F(1,k).
(7.29)
Thus, according to (7.29) the solution to the problem is reduced to obtain
ing the expression for the function F(1,k). The latter is the expression for the
(1,k)exactness of the optimal (1,k,0)algorithm. Remember that the (1,k,0)
algorithm consists of 1 step. This step is to enclose the k IE’s at the points of the
line segment AB. It is easy to prove that in this case the optimal solution is to
divide the line segment AB into (k+1) equal parts. It follows from this consid
eration that the (1,k)exactness of the optimal (1,k,0)algorithm is equal to
F(1,k)=k+1.
(7.30)
By substituting (7.30) into (7.29), we obtain the expression for the function
(7.28) in the explicit form:
(7.31)
F(n,k)=(k+1)n .
The expression (7.31) allows one to formulate the essence of the optimal (n,k,0)
algorithm. Each step of the algorithm is to divide the initial “indeterminacy inter
val” (the line segment AB) and all the next “indeterminacy intervals” into (k+1)
equal parts. As a result of the application of the algorithm, we obtain the formula
(7.31), which characterizes the effectiveness of the optimal (n,k,0)algorithm.
7.7.3. Special Cases of the Optimal (n,k,0)algorithm
Consider some special cases of the optimal (n,k,0)algorithm. For the
case k=1 the formula (7.31) is reduced to the formula (7.17), which expresses
Alexey Stakhov
THE MATHEMATICS OF HARMONY
392
the (n,1)exactness of the “binary” algorithm. For the case n=1 the formula (7.31)
is reduced to the formula (7.19), which expresses the (1,k)exactness of the “rul
er” algorithm. It follows from this examination that the wellknown measure
ment algorithms, “binary” algorithm and “ruler” algorithm, are partial cases of
the optimal (n,k,0)algorithm.
Similar to the “binary” algorithm, which generates the “binary” number
system (7.20), the optimal (n,k,0)algorithms generate the positional number
systems with radix R=k+1, i.e.
(7.32)
A=an1Rn1+ an2Rn2+…+aiRi+…+ a1R1+ a0R0,
where ai∈{0,1,2,3,…,k} is a numeral of the ith digit.
The expression (7.32) has the following “measurement” interpretation. The
numeral an1 in (7.32) encodes the “indications” of the IE’s on the first step of the
optimal (n,k,0)algorithm according to the rule:
0 = 000...00
1 = 100...00
2 = 110...00
...
(7.33)
k − 1 = 111...10
k = 111...11
The numeral an2 encodes the “indications” of the IE’s on the second step of
the algorithm and the numerals a1 and a0 encode the “indications” of the k IE’s
according to the rule (7.33) on the (n1)th and nth steps of the optimal
(n,k,0)algorithm, respectively.
It is clear that for k=9 the optimal (n,9,0)algorithm generates the deci
mal system and for k=59 the Babylonian sexagesimal system. It follows from
this consideration that the optimal (n,k,0)algorithms generate all historical
ly wellknown positional systems.
7.8. Optimal Measurement Algorithms Based on the Arithmetical Square
7.8.1. The Optimal (n,k,1)algorithms
Consider the class of the (n,k,1)algorithms. Remember that the restriction
S≡1 means that the indicator elements (IE) are moving along the line segment
AB only in the direction from point A to point B. This means, if any IE on any step
is found to the right of the “unknown” point X, then this IE will “leave the field,”
Chapter 7
Algorithmic Measurement Theory
393
i.e. this IE cannot be used for further measurement. Note that the above “count
ing” algorithm (Fig. 7.6) is the simplest example of the restriction S≡1.
Let the first step of the optimal ( n, k,1) algorithm be to enclose the k IE’s
to the certain points C1,C2,…,Cj,Cj+1, …, Ck of the line segment AB (Fig. 7.8).
Consider the situations (7.24) that can appear after the first step of the algo
rithm. Consider the situation
X∈C jCj+1.
(7.34)
It is clear for the situation (7.34) that the j IE’s are on the left of the “un
known” point X and the rest, the (kj)th IE’s, are on the right of point X.
In accordance with the restriction S≡1, the (k j) IE’s, which are on the
right of the “unknown” point X, cannot be used for further measuring. This
means that the second step of the algorithm is to enclose to the line segment
CjCj+1 only those j IE’s which are found on the left of the “unknown” point X
after the first step of the algorithm. That is why we can only use the optimal (n
1,j,1)algorithm for the situation (7.34). In accordance with the inductive as
sumption, the optimal (n1,j,1)algorithm can divide the line segment CjCj+1
into F(n1,j) equal parts ∆=1, that is,
CjCj+1=F(n1,j).
Consider the situations
(7.35)
X∈AC1
and
(7.36)
(7.37)
X∈CkB.
For the situation (7.36) all IE’s are to the right of the point X after the first
step. This means that all IE’s “leave the field” and the process of the measure
ment is over. It is clear from this examination that the line segment AC1 must be
the line segment of a single length, i.e.
AC1=1.
(7.38)
For the situation (7.37) all IE’s are to the left of point X after the first step of
the algorithm and they can be enclosed to the points of the line segment CkB on
the second step of the algorithm. Using the optimal (n1,k,1)algorithm, we can
divide the line segment CkB into F(n1,k) equal parts ∆=1, i.e.
CkB=F(n1,k).
(7.39)
Using the expressions (7.26), (7.35), (7.38), (7.39), we obtain the follow
ing recursive relation for the calculation of the function F(n,k), which gives the
(n,k)exactness of the optimal (n,k,1)algorithm:
F(n,k)=1+F(n1,1)+F(n1,1)+F(n1,2)+…+F(n1,j)+…+F(n1,k1)+F(n1,k). (7.40)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
394
Using the recursive relation (7.40), we can write the following recursive for
mula for the calculation of the function F(n, k1):
F(n,k1)=1+F(n1,1)+F(n1,1)+F(n1,2)+…+F(n1,j)+…+F(n1,k1). (7.41)
Comparing the expressions (7.40) and (7.41), we can rewrite the recursive
relation (7.40) in the following form:
F(n,k)= F(n,k1)+ F(n1,k1).
(7.42)
Calculating the values of the function F(n, k) for the special cases n=0 and
k=0, we clearly have:
F(0,k)=1
(7.43)
F(n,0)=1.
(7.44)
For an explanation of the “physical sense” of the expressions (7.43) and (7.44)
we can examine the optimal (n, k,1)algorithm for cases n=0 and k=0. The case
n=0 means that we do not have any step for the measurement and, hence, the
“indeterminacy interval” (the line segment AB) cannot be decreased. Thus, the
expression (7.43) is valid.
Now, suppose that k=0. This means that we do not have any “Indicator
Elements” for the measurement and the “indeterminacy interval” (the line seg
ment AB) cannot be decreased. Thus, the expression (7.44) is valid.
7.8.2. Arithmetical Square
Using the recursive relation (7.42) with the initial terms (7.43) and (7.44),
we can construct a table of the values of the function F(n,k) (Table 7.1). We
can see that Table 7.1 coincides with the wellknown Arithmetical Square or
Pascal Triangle and that the terms F ( n, k ) are Binomial Coefficients, that is,
F ( n, k ) = C nk+ k = C nn+ k .
(7.45)
Table 7.1. Arithmetical Square
k/n
0
1
0
1
1
1
1
2
2
1
3
3
1
4
4
1
5
5
1
6
...
...
...
2
1
3
6
10
15
21
...
n
1
F ( n, 1)
F ( n, 2 )
F ( n, 3 )
3
1
4
10
20
35
56
...
4
1
5
15
35
70
126
...
5
1
6
21
56
126
252
...
F ( n, 5 )
#
k
#
1
#
F (1, k )
#
F ( 2, k )
#
F ( 3, k )
#
F ( 4, k )
#
F ( 5, k )
###
...
#
F ( n, k )
F ( n, 4 )
Chapter 7
Algorithmic Measurement Theory
395
7.8.3. The Optimal Measurement Algorithms Based on Arithmetical
Square
The arithmetical square (Table 7.1) allows to demonstrate the optimal
(n,k,1)algorithm. In fact, for the given n and k, the value of the function F(n,k)
is at the intersection of the nth column and kth row of the arithmetical square.
The coordinates of the k IE’s, which should be enclosed to the points
C1,C2,…,Cj,Cj+1, …, Ck of the initial line segment AB, are in the nth column of
the arithmetical square, that is,
AC1 = 1, AC2 = F ( n,1) , ..., AC j = F ( n, j − 1) , AC j +1
(7.46)
= F ( n, j ) , ..., AC k = F ( n, k − 1) .
If after the first step of the algorithm the j IE’s are to the left of point X and the
rest (kj) IE’s are to the right of X, the “indeterminacy interval” about X is dimin
ished up to the line segment CjCj+1. The length of the latter is equal to the binomi
al coefficient F(n,j). This binomial coefficient is at the intersection of the (n1)th
column and the jth row of the arithmetical square. To obtain the binomial coeffi
cient F(n,j) we have to move from the initial coefficient F(n,k) by one column to
the left and by (kj) rows upwards. On the second step we have to accept the
point Cj as the new beginning of coordinates. Here the second step is to enclose j
IE’s to the points D1, D2, …, Dj of the new “indeterminacy interval” CjCj+1. The
coordinates of the points D1, D2, …, Dj with respect to the point Cj the new begin
ning of coordinates are in the (n1)th column of the arithmetical square, that is,
C j D1 = 1, C j D2 = F ( n − 1,1) , C j D3 = F ( n − 1, 2) , ...,C j D j = F ( n − 1, j − 1) .
(7.47)
This process continues up to the exhaustion either of all measurement steps
or all IE’s.
7.8.4. The Example of the Optimal (n, k, 1)algorithm
Consider the optimal (3, 3, 1)algorithm (Fig. 7.9). The algorithm con
sists of 3 steps (n steps in the general case) and uses 3 IE’s (k IE’s in the
general case). The value of the initial “indeterminacy interval” is the binomi
al coefficient F(3, 3)=20, which is at the intersection of the 3d column and 3
d row of the arithmetical square.
The first step of the algorithm at line segment [0,20] is to enclose 3 IE’s to
the points 1, 4, 10 (Fig. 7.9). In accordance with the IE“indications,” four
situations may appear after the first step: [0,1], [1,4], [4,10], [10,20].
The second step:
(1a) For the situation [0,1] the process of measurement is over because all
IE’s are on the right of the point X.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
396
0 1
4
10
20
(1b) For the situa
tion [1,4] we have the
0 1
only 1 IE, which can be (1a)
1 2
4
enclosed at point 2.
(1b)
(1c) For the situa
7
10
4 5
tion [4,10] we have 2 (1c)
16
10 11
13
20
IE’s, which can be en (1d)
closed to points 5 and 7.
Figure 7.9. The optimal (3, 3, 1)algorithm
(1d) For the situa
tion [10,20] we have 3 IE’s. For this case the coordinates of three points of the
IE’s enclosing line segment [10,20] with respect to the point 10, the new begin
ning of coordinates, are in the second column of the arithmetical square (Table
7.1) above binomial coefficient 10. Those are the binomial coefficients 1, 3, 6. By
summing these numbers with the number 10, we obtain the coordinates of the
points of the IE’s enclosed on the second step, namely: 11=10+1, 13=10+3, and
16=10+6.
The third step. After the second step, the following situations will appear: [1,2],
[2,4], [4,5], [5,7], [7,10], [10,11], [11,13], [13,16], [16,20]. Note that for the situa
tions [1,2], [4,5], [10,11] the measurement process is over on the second step.
It is clear from Fig. 7.9 that the optimal (3,3,1)algorithm can divide the
initial line segment [0,20] into 20 equal parts.
7.9. Fibonacci Measurement Algorithms
7.9.1. The Optimal Measurement Algorithms Based on the Fibonacci
pnumbers
Return to the “indicator” measurement model (Fig. 7.4) and try to intro
duce the above Asymmetry Principle of Measurement into this model. The
following “restriction” to the movement of the IE along the line segment AB
follows from the Asymmetry Principle of Measurement.
Suppose that the IE is enclosed at the point C on the first step of the n
step measurement algorithm (Fig. 7.10).
Thereafter, two situations (a) and (b) appear after the first step, as shown
in Fig. 7.10. It is clear that for the situation (a) we can enclose the IE at any
point of the “indeterminacy interval” CB on the next step. For the situation (b)
Chapter 7
Algorithmic Measurement Theory
397
B
C
A
we do not have the right to en
close the IE to the points of the
“indeterminacy interval” AC, be
B
C
cause on the first step the balance (a)
F(n1)
moves to the opposite position
A
C
and we need p units of discrete (b)
F(np1)
time for returning the balance
Figure 7.10. The first step of the Fibonacci
into the initial position. Thus,
measurement algorithm
the restriction S, which follows
from the Asymmetry Principle of Measurement, consists of the fact that for the
situation (b) it is forbidden to enclose the IE to the points of the line segments
AC during the next p steps of the algorithm.
We can use the above restriction for synthesis of the optimal measurement
algorithm. Here we introduce the following “inductive assumption.” Suppose that
for the given n≥0 and p≥0 there is the optimal nstep measurement algorithm,
which divides the initial line segment AB into Fp(n) equal parts of ∆=1, that is,
(7.48)
AB=Fp(n).
Suppose that the first step of the algorithm is to enclose the IE at the
point C (Fig. 7.10). Then, after the first step two situations can appear. Line
segment CB is the “indeterminacy interval” for situation (a). According to
the restriction S we can enclose the IE at any point of CB on the second step.
By using the optimal (n1)step algorithm in this situation we can divide the
line segment CB into the Fp(n1) equal parts ∆=1 and, hence,
CB = Fp(n1).
(7.49)
Line segment AC is the “indeterminacy interval” for the situation (b).
According to the restriction S, we cannot enclose the IE to the points of the
line segment AC during the next p steps of the algorithm. Let us consider the
situation (b) for two possible cases: (1) p≥n1 (2) p<n1.
It is clear that all steps of the algorithm for case (1) will be “exhausted”
much earlier before the “prohibition” to enclose the IE to the points of the
line segment AC will be taken off. This means that for situation (1) the mea
surement is over after the first step. That is why line segment AC should be
the line segment of unit length, that is,
AC=1.
(7.50)
Consider the situation (b) for case (2). For this case we have the right to
enclose the IE to the points of the line segment AC only after p steps. Then,
acting by the optimal (np1)algorithm we can divide the line segment AC
into the Fp(np1) equal parts D=1, that is,
Alexey Stakhov
THE MATHEMATICS OF HARMONY
398
AC=Fp(np1).
(7.51)
Taking into consideration that AB=AC+CB, we can write the following
expression for the function Fp(n) for cases (1) and (2), respectively:
Fp(n)= Fp(n1)+1 with p≥n1
(7.52)
Fp(n)= Fp(n1)+ Fp(np1) with p<n1.
(7.53)
Note that the recursive formula (7.53) coincides with the recursive for
mula for the Fibonacci pnumbers introduced in Chapter 4. That is why the
measurement algorithms that correspond to the mathematical formulas
(7.52) and (7.53) were called Fibonacci Measurement Algorithms [20]. Re
member that Fibonacci pnumbers is a numerical sequence that is given by
the following recursive relation
Fp(n)=Fp(n1)+ Fp(np1)
at the seeds:
(7.54)
(7.55)
Fp(1)= Fp(2)=…= Fp(p+1)=1.
Consider in greater detail the mathematical formulas (7.52) and (7.53). For
the case p≥n1 the Fibonacci measurement algorithms are described by the for
mula (7.53). The recursive relation (7.52) may be expressed in the explicit
form. In fact, by decomposing the function Fp(n1) in (7.52) and all the fol
lowing functions Fp(n2), Fp(n3), Fp(1) in accordance with the same recur
sive relation (7.52), we obtain the following expression:
Fp(n)=n+Fp(0).
(7.56)
It follows from the “physical” reasoning that Fp(0)=1 and then the expres
sion (7.56) may be represented in the following “explicit” form:
Fp(n)=n+1.
(7.57)
Note that the formula (7.57) generates the natural numbers.
By joining the formulas (7.57) and (7.53), we can write the following
expression for the function Fp(n)
with p ≥ n − 1
n = 1
Fp (n) = 
.
(7.58)
 Fp ( n − 1) + Fp ( n − p − 1) with p < n − 1
Consider some special cases of the formula (7.58) for the different values of p.
Let p=0. For this case the formula (7.58) is reduced to the following:
F0(n)=2 F0(n1)
(7.59)
F0(0)=1.
(7.60)
It is clear that the recursive formula (7.59) at seed (7.60) generates the
“binary” sequence:
Chapter 7
Algorithmic Measurement Theory
399
1, 2. 4. 8, 16, …, F0(n)=2n1.
It is clear that the Fibonacci measurement algorithm that corresponds to
this case is reduced to the classical “binary” algorithm.
Let p=∞. This means that the IE “leaves the field” when it is on the right
of the point X. For this case the formula (7.58) takes the form of the expres
sion (7.57) that generates natural numbers. It is clear that the classical “count
ing” algorithm is the optimal Fibonacci measurement algorithm correspond
ing to this case. Table 7.2 gives the values of the (n,1)exactness (7.58) of the
optimal Fibonacci algorithms for the different values p.
The expression (7.58)
Table 7.2. The (n,1)exactness of the Fibonacci
gives
the (n,1)exactness
measurement algorithms
of the optimal nstep Fi
bonacci algorithm for the
Fp ( n ) / n 1
2
3
4
5
6
7
8
9
F0 ( n )
2
4
8
16 32 64 128 256 512 restriction S that follows
from the Asymmetry Prin
F1 ( n )
2
3
5
8
13 21 34 55 89
ciple of Measurement. We
F2 ( n )
2
3
4
6
9
13 19 28 41
find the expression for the
F3 ( n )
2
3
4
5
7
10 14 19 26
standard weights for Fi
#
#
#
#
#
#
#
#
#
#
bonacci algorithms in the
8
9
10
F∞ ( n )
2
3
4
5
6
7
form:
{Wp(1), Wp(2), …, Wp(i), …, Wp(n)},
where Wp(i) is the ith standard weight. It is easily proven that the standard
weight Wp(n) is given by the following expression:
Wp(n)= Wp(n1)+Wp(np1) with
n>p+1
(7.61)
Wp(1)= Wp(2)= … = Wp(p+1)=1.
(7.62)
Table 7.3 gives different variants of the standard weight systems correspond
ing to different values of p.
A comparison of the expressions (7.61) and (7.62) with expressions (7.52)
and (7.53) shows that
Table 7.3. Standard weights for the Fibonacci
the standard weights
measurement algorithms
Wp(n) for Fibonacci’s
Wp ( n ) / n 1
2
3
4
5
6
7
8
9
measurement algo
W0 ( n )
1
2
4
8
16 32 64 128 256
rithms coincide with
W1 ( n )
1
1
2
3
5
8
13 21 34
the pFibonacci num
W2 ( n )
1
1
1
2
3
4
6
9
13
bers [20], in particular,
W3 ( n )
1
1
1
1
2
3
4
5
7
with the classical Fi
#
#
#
#
#
#
#
#
#
#
bonacci numbers for
W
n
1
1
1
1
1
1
1
1
1
(
)
∞
the case p = 1 .
Alexey Stakhov
THE MATHEMATICS OF HARMONY
400
7.9.2. The Example of the Fibonacci Measurement Algorithm
Here we examine the optimal Fibonacci’s measurement algorithm given by
the expression (7.58) for the case p=1. Let n=5. Consider the 5step Fibonacci
algorithm (Fig. 7.11) corresponding to the case p=1. It follows from Table 7.2
that the above 5step Fibonacci algorithm provides the (5,1)exactness F1(5)=13.
This means that the optimal 5step Fibonacci algorithm divides the line segment
[0,13] into 13 equal parts. Thus, the given Fibonacci measurement algorithm
consists of 5 steps and uses 5 standard weights {1, 1, 2, 3, 5} (see Table 7.3).
Let us consider the first 3 steps of the given algorithm.
The first step. We can use the first standard weight 5 for comparison with
the measurable weight. This means that in the terms of the “indicator” model
we have to enclose the IE at the point 5 of the line segment [0,13] (Fig. 7.11).
We can see that the first step consists in the division of the line segment
[0,13] in the Fibonacci ratio: 13=5+8. Two situations (a) and (b) (Fig. 7.11)
can appear after the first step.
0
5
(a)
(b)
13
5
0
8
13
5
forbidden
8
(c)
5
(d)
10
13
10
13
8
forbidden
(e)
0
2
5
(f)
8
(g)
10
forbidden
2
(h)
(i)
0
5
2
Figure 7.11. Example of the Fibonacci measurement algorithm
The second step. For situation (a) we can use the next standard weight 3 and
divide the line segment [5,13] by the IE in the Fibonacci ratio: 8=3+5. Two new
situations (c) and (d) (Fig. 7.11) appear after the second step.
Chapter 7
Algorithmic Measurement Theory
401
For the situation (b) the second step is “empty” because in accordance with
restriction S it is forbidden to enclose the IE to the points of the line segment [0,
5] on the second step.
The third step. For the situation (c) we can use the next standard weight
2 to divide the line segment [8,13] in the Fibonacci ratio: 5=3+2. Two new
situations (f) and (g) appear after the third step.
For the situation (e) we can return to the situation (b) on the third step.
In accordance with the restriction S we can enclose the IE at any point of the
line segment [0,5] on the third step. We can use the standard weight 2 to
divide the line segment [0,5] in the Fibonacci ratio: 5=2+3. Two new situa
tions (h) and (i) appear after the third step.
It is easy to trace the actions of the Fibonacci algorithm for the next two
steps (Fig. 7.11).
We can see from this example that the essence of the Fibonacci measure
ment algorithm is to divide the “indeterminacy interval” obtained on the pre
ceding step in the Fibonacci ratio. It is easy to show that this general principle
is valid for any arbitrary p. The division of the “indeterminacy interval” for this
case is affected according to the recursive relation for Fibonacci pnumbers.
7.10. The Main Result of Algorithmic Measurement Theory
7.10.1. A Further Generalization of the BachetMendeleev Problem
We can give a further generalization of the BachetMendeleev problem
[20]. Suppose that we use k balances simultaneously (k is a natural number, 1,
2, 3 …). We can imagine that one and the same measurable weight Q is on the
lefthand cups of all balances. Such a situation arises for the case of the “par
allel measurement” of one and the same measured magnitude Q, when we
compare the measurable magnitude Q with the standard weights by means of
k “comparators” (such situation is widely used in the measurement of electric
magnitudes). In this case the generalized BachetMendeleev problem can be
formulated as follows: it is necessary to synthesize the optimal nstep mea
surement algorithm to determine the value of the measurable magnitude Q
with the help of k balances (“comparators”), which have the “inertness” р, for
the condition that on each step the standard weights can be placed on the free
cups of those balances, which are in the initial state.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
402
Suppose that all balances, which participate in measuring, have the “inert
ness” p (p=0, 1, 2, 3, …). This results in the problem to find the “parallel” optimal
(n,k,S)algorithm, which uses k balances having the “inertness” p. For the simula
tion of the “inertness” of the balances (or “Indicator Elements”) we can introduce
the notion of the state of the jth IE on the lth step (j=1, 2, 3, …, k; l=1, 2, 3, …, n).
Denote the latter by pj(l). It follows from the physical examination of the Bachet
Mendeleev problem that the integer function pj(l) has the following properties:
0≤ pj(l)≤p
(7.63)
pj(l+1)= pj(l)1.
(7.64)
We can clarify the “physical sense” of the expressions (7.63) and (7.64). Note
that the case pj(l)=0 means that the jth balance is in the initial position (Fig. 7.3
a) and the case pj(l)=p means that the balance is in the opposite position (Fig. 7.3
b). The expression (7.64) depicts the process of returning the jth balance from
the opposite position (Fig. 7.3b) to the initial position (Fig. 7.3a). According to
(7.64) the state of the IE is decreased by 1 on the next step. Thus, if the jth IE on
the lth step of the algorithm turns out to be in the state pj(l)=p, then its states on
the next steps will decrease successively according to the following:
pj(l)=p→ pj(l+1)=p1→ pj(l+2)=p2→ … → pj(l+p1)=1→ pj(l+p)=0.
Note that the case of pj(l)=0 means that the jth IE on the lth step of the
algorithm may be enclosed to the points of the “indeterminacy interval.”
After such preliminary remarks we can try to synthesize the optimal (n,k,S)
algorithm. Before the lth step is carried out, we renumber all indicator ele
ments so that their states would be arranged according to the rule:
pk(l)≥ pk1(l)≥ pk2(l)≥ … ≥p2(l)≥ p1(l),
(7.65)
where pj(l) is the state of the jth IE on the lth step (j=1,2,3, …, k; l=1,2,3, …, n).
We denote the IEstates on the first step of the (n,k,S)algorithm by p1, p2,
p3, …, pk. The initial IEstates are disposed in accordance with (7.65), that is,
pk≥ pk1≥ pk2≥…≥ p2≥ p1.
(7.66)
7.10.2. The Main Recursive Relation of the Algorithmic Measurement
Theory
We denote the (n,k)exactness of the optimal (n,k,S)algorithm for this case by
F(n,k)=Fp(n; p1, p2, p3, …, pk).
Let the initial IEstates p1, p2, p3, …, pk be disposed according to (57) so
that the first t states p1, p2, p3, …, pt are equal to 0, that is,
Chapter 7
Algorithmic Measurement Theory
403
p1=p2= p3= …= pt =0.
(7.67)
This means that only the first t IE’s, which satisfy (7.66), can be enclosed to
the points of the line segment AB on the first step of the (n, k, S)algorithm.
Suppose there is the optimal (n, k, S)algorithm, which for the above restric


, 0,..., 0 , pt +1 , pt + 2 , ..., pk  equal parts of
tion divides the line segment AB into Fp  n; 0N


t
∆=1. Let the first step of the algorithm be to enclose the t IE’s to the points C1,
C2, …, Ct (Fig. 7.12).
A
C1
C2
Cj
Cj+1
...
1
2
Ct
B
...
j+1
t+1
Figure 7.12. Synthesis of the optimal (n, k, S)algorithm
for the general case
After the first step of the algorithm, the (t+1) situations
X∈AC1, X∈ C1C2, …, X∈CjCj+1, …, X∈CtB
appear dependent upon the “indications” of the t IE’s.
Consider the situation X∈ΑC1. For this situation all t IE’s are on the right of
the “unknown” point X and hence they turn into the states of p after the first
step. In accordance with property (7.64) the rest of the (kt) IE’s decrease their
states by 1 on the second step, that is, their states are:
pt+11, pt+21, …, pk1.
If we arrange all k IE’s in accordance with (7.65), we obtain the following IE
states on the second step of the algorithm:
pt +1 − 1, pt + 2 − 1, ..., pk − 1, p, p, ..., p .
(7.68)
t
If we use the optimal (n1,k,S)algorithm with the initial IEstates (7.68)
for this situation, then in accordance with the inductive assumption we can
divide the line segment AC1 into


F p  n − 1; pt +1 − 1, pt + 2 − 1, ..., pk − 1, p, p, ..., p 


t


equal parts of ∆ = 1.
Consider the situation X∈C1C2. It follows from Fig. 7.12 that for this case
the j IE’s are on the left of the point X and hence remain in the 0states, the
(tj) IE’s turn into the p states and the rest of the (kt) IE’s decrease their
states by 1, that is,
pt+11, pt+21, …, pk1.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
404
If we arrange all k IE’s in accordance with (7.65), we obtain the following
sequence of the IEstates on the second step of the algorithm:
0, 0, ..., 0 , pt +1 − 1, pt + 2 − 1, ..., pk − 1, p, p, ..., p .
(7.69)
j
t− j
If we use the optimal (n1,k,S)algorithm with the initial IEstates (7.69)
for this situation, then in accordance with the inductive assumption we divide
the line segment CjCj+1 into


F p  n − 1; 0, 0, ..., 0 , pt +1 − 1, pt + 2 − 1, ..., pk − 1, p, p, ..., p 


j
t− j


equal parts of ∆ = 1.
At least, it is easy to show that for situation X∈CtB the optimal (n1,k,S)
algorithm divides the line segment CtB into


F p  n − 1; 0, 0, ..., 0 , pt +1 − 1, pt + 2 − 1, ..., pk − 1


t


equal parts of ∆ = 1.
Taking into consideration the evidence equality
AB = AC1 + C1C2 + ... + C jC j +1 + ... + Ct B,
we obtain the following recursive relation for the calculation of (n,k)exactness
of the optimal (n,k,S)algorithm:


F ( n, k ) = Fp  n; 0, 0, ..., 0 , pt +1, pt +2 , ..., pk 
 
t




(7.70)
Fp  n − 1; 0, 0, ..., 0 , pt +1 − 1, pt +2 − 1, ..., pk − 1, p, p, ..., p .


j =0
j
t− j


Define the initial conditions for the calculation of Fp(n,k) according to the
recursive relation (7.70). We take into consideration that for the cases n=1 and
p1=p2=p3=…=pt=0, pt+1>0 the optimal (1,k,S)algorithm divides the line segment
AB into t=1 equal parts, that is,


F p  1; 0, 0, ..., 0 , pt +1 , pt + 2 , ..., pk  = t + 1.
(7.71)
 
t


Let us introduce the following definition:
0, with n < 0
F p ( n, k ) = 
.
(7.72)
1, with n = 0
It follows from the “physical sense” of the function F(n,k)=Fp(n; p1, p2, p3, …,
pk) that for the case n≥p1>0 we have:
=
t
∑
Chapter 7
Algorithmic Measurement Theory
405
Fp(n; p1, p2, p3, …, pk)=Fp(np1; 0, p2 p1, p3p1, …, pkp1).
(7.73)
The recursive relation (7.70) at the initial terms (7.71) and (7.72) is the main
result of the algorithmic measurement theory. This recursive relation gives a the
oretically infinite set of optimal measurement algorithms for the given n, k and p.
7.10.3. The Unusual Results
The recursive relation (7.70) at the initial terms (7.71) and (7.72) contains a
number of remarkable formulas of discrete mathematics.
Consider now the case of p=0. This means that the IEstates are equal to 0
on each step of the algorithm, that is,
(7.74)
pk(l)= pk1(l)= … = p2(l)= p1(l)=0.
Taking into consideration (7.74), we can prove that the recursive relation
(7.70) and the initial condition (7.71) are reduced to the following:




F p  n; 0, 0, ..., 0  = ( k + 1) F p  n − 1; 0, 0, ..., 0 
(7.75)
 


k
k






F p  1; 0, 0, ..., 0  = ( k + 1) .
(7.76)
 
k


If we use the following definition


F p  n; 0, 0, ..., 0  = F ( n, k ) ,
 
k


we can then represent the expressions (7.75) and (7.76) as follows:
F(n,k)=(k+1)F(n1,k)
(7.77)
F(1,k)=k+1.
(7.78)
We can see that the expressions (7.77) and (7.78) coincide with the formu
las (7.28) and (7.30) for the optimal (n,k,0)algorithm. As is shown above, the
recursive function F(n,k) that is defined by the recursive relation (7.77) at the
seeds (7.78) can be expressed in the explicit form given by the formula (7.31).
It follows from this consideration that for the case p=0 the general opti
mal (n,k,S)algorithm given by the formulas (7.70) and (7.71) is reduced to
the optimal (n, k, 0)algorithm.
Now, let us consider the case when
p=∞
and
p1=p2= p3= …= pk =0.
(7.79)
(7.80)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
406
The “physical sense” of the conditions (7.78) and (7.80) are as follows. The
condition (7.80) means that all k IE’s are in the 0states on the first step of the
algorithm. According to the condition (7.79) every IE that is found on the right
above the “unknown” point X at any step of the algorithm “leaves the field,” that
is, cannot participate in further measurement.
Let us consider the function


F p  n; 0, 0, ..., 0 , p, p, ..., p  .
(7.81)
 
j
k− j


It follows from the “physical sense” that for the cases (7.79) and (7.80) the
function (7.81) cannot depend on the (kj) IE’s that are found in the state p=∞.
Hence, for this case the function (7.81) depends only on the number n of the
algorithm steps and the number j of the IE’s that are found to the right of the
point X on the first step of the algorithm. These “physical” considerations
can be expressed in the following form:


F p  n; 0, 0, ..., 0 , p, p, ..., p  = F ( n, j ) .
(7.82)
 
j
k− j


Note that


F p  n; p, p, ..., p  = F ( n, 0 ) = 1.
 
k


(7.83)
The “physical sense” of the expression (7.83) consists of the fact that all IE’s
are found in the state p=∞ and we therefore do not have the possibility of de
creasing the “indeterminacy interval.”
Now let us return to formulas (7.70) and (7.71). It is clear that for the condi
tions (7.79) and (7.80) the formulas (7.82) and (7.83) take the following forms,
respectively:


F p  n; 0, 0, ..., 0  = F ( n, k )
(7.84)
 
k




F p  1; 0, 0, ..., 0  = k + 1.
 
k


(7.85)
Taking into consideration definitions (7.82) and (7.83), we can represent
the expressions (7.70) and (7.71) as follows:
F(n,k)=1+ F(n1,k)+F(n1,1)+F(n1,2)+…+ F(n1,j)+…+ F(n1,k)
(7.86)
Chapter 7
Algorithmic Measurement Theory
407
F(1,k)=k+1.
(7.87)
Comparing the expressions (7.86) and (7.87) with the corresponding recur
sive relations for the optimal (n,k,1)algorithm, we can see that these expres
sions coincide and hence,
F ( n, k ) = C nk+ k = C nn+ k .
It follows from this consideration that for conditions (7.79) and (7.80) the
general optimal (n,k,S)algorithm given by formulas (7.70) and (7.71) is reduced
to the optimal (n,k,1)algorithm based on the arithmetical square.
At least, consider the case
k=1, p≥0 and p1=0.
(7.88)
This means that we deal with the (n,1,S)algorithm (k=1) and the IEstate
on the first step of the algorithm equal to 0. Then, the expressions (7.70) and
(7.71) for this case take the following form:
Fp(n;0)= Fp(n1;0)+Fp(n1;p)
(7.89)
(7.90)
Fp(1;0)=2.
Let us consider the function Fp(n1;p) in (7.89). Using (7.73), we can write
the following expression:
Fp(n1;p)= Fp(np1;0).
Then, the expression (7.89) can be written as follows:
(7.91)
Fp(n;0)= Fp(n1;0)+Fp(np1;0).
Next let us introduce the following definition:
(7.92)
Fp(n;0)= Fp(n).
Then, the expressions (7.92) and (7.90) can be written as follows:
Fp(n)= Fp(n1)+Fp(np1)
(7.93)
Fp(1)=2.
(7.94)
Comparing the expressions (7.93) and (7.94) with the corresponding recur
sive relations for the optimal Fibonacci measurement algorithms, we can see that
these expressions coincide and, hence, for the case (7.88) the general optimal
(n,k,S)algorithm given by the expressions (7.70) and (7.71) are reduced to the
Fibonacci measurement algorithm.
The main result of the algorithmic measurement theory together with all
the unusual results is demonstrated in Fig. 7.13.
Thus, the “unexpectedness” of the main result of the algorithmic measure
ment theory (Fig. 7.13) consists of the following. The general recursive rela
tion (7.70) at the seed (7.71) sets in general form an infinite number of new,
Alexey Stakhov
THE MATHEMATICS OF HARMONY
408
p=0
k ≥1
( k + 1)
0≤ p≤∞
n
←
↓
k =1
2
n
Binary
numbers
Fp ( n , k )
p=∞
→
↓
←
Fp ( n ) = Fp ( n − 1) + Fp ( n − 2 )
Fibonacci
p − numbers
C nk+ k = C nn+ k
Binomial
coefficients
↓
→
n +1
Natural
numbers
Figure 7.13. The main result of the algorithmic measurement theory
until now unknown, optimal measurement algorithms. All wellknown classical
measurement algorithms that are used in measurement practice (the “binary” al
gorithm, the “counting” algorithm, the “ruler” algorithm) are special limiting cas
es of the optimal measurement algorithms. The main recursive relation (7.70) at
the seed (7.71) includes a number of the wellknown combinatorial formulas as
special cases, in particular, the formula (k+1)n, the formula C nk+ k = C nn+ k that gives
binomial coefficients, the Fibonacci recursive relation, and finally the formulas for
the “binary” (2n) and natural (n+1) numbers.
7.11. Mathematical Theories Isomorphic to Algorithmic Measurement
Theory
7.11.1. What is an Isomorphism?
The concept of Isomorphism [from the Greek words isos (equal), and morphe
(shape)] is used widely in mathematics. Informally, isomorphism is a kind of
correspondence between objects that show a relationship between two proper
ties or operations. If there is isomorphism between two structures, we call these
structures isomorphic. The word “isomorphism” applies when two complex
structures can be reflected onto each other in such a way that each part of
one structure fits a corresponding part of the other structure.
We can give the following physical analogies of isomorphism:
1. A solid wood cube and a solid metallic cube are both solid cubes; although
their physical nature differs, their geometric structures are isomorphic.
2. The Big Ben Clock in London and a wristwatch; although the clocks
differ greatly in size, their timecounting mechanisms are isomorphic.
Chapter 7
Algorithmic Measurement Theory
409
We can give different examples of isomorphism in mathematics. In abstract
algebra, two basic isomorphisms are defined: Group Isomorphism, the isomorphism
between groups, and Ring Isomorphism, the isomorphism between rings. There is
also Graph Isomorphism in graph theory. In linear algebra, the isomorphism can
also be defined as a Linear Map between two vector spaces.
7.11.2. Isomorphism between the “Balance” and “Rabbits”
In the above we formulated the Asymmetry Principle of Measurement, the main
methodological principle of algorithmic measurement theory. Note that this principle
has practical origin because it reflects some essential properties of the comparators
used in AnalogtoDigit Converters. For the first time, the isomorphism between bal
ances and comparators was stated in this author’s Doctor of Science dissertation Syn
thesis of Optimal Algorithms of AnalogtoDigit Conversion (1972).
The above Asymmetry Principle of Measurement is based on the concept of
the “inertial balances” used for measurement. If we define by I the initial posi
tion of the balance (Fig. 7.3a) and by O the opposite position (Fig. 7.3b), then
the functioning of the balance can be described by using two transitions:
I
I →
(7.95)
O
(7.96)
O → I.
The transition (7.95) means that the balance can be in one of two extreme
positions, I or O, after we place the next standard weight on the free cup of the
balance. The transition (7.96) means that if the balance is in the position O,
then during a certain time the balance is coming back into the initial position I.
In Chapter 2, we described Fibonacci’s problem of rabbit reproduction.
Let us recall that the “Law of rabbit reproduction” boils down to the follow
ing rule. Each mature rabbit’s pair А gives birth to a newborn rabbit pair B
during one month. The newborn rabbit’s pair becomes mature during one
month and then in the following month said pair starts to give birth to one
rabbit pair each month. Thus, the maturing of the newborn rabbits, that is,
their transformation into a mature pair is performed in 1 month. We can
model the process of “rabbit reproduction” by using two transitions:
A → AB
(7.97)
(7.98)
B → A.
Note that the transition (7.97) simulates the process of the birth of the new
born rabbit pair B and the transition (7.98) simulates the process of the maturing
Alexey Stakhov
THE MATHEMATICS OF HARMONY
410
of the newborn rabbit pair B. The transition (7.97) reflects an asymmetry of rabbit
reproduction because the mature rabbit pair А is transformed into two nonidenti
cal pairs, the mature rabbit pair А and the newborn rabbit pair B.
Comparing the transitions (7.95) and (7.96) with the transitions (7.97) and
(7.98), we can see analogies or isomorphism between them. Moreover, this
isomorphism is confirmed by the fact that the solutions of these problems come
to one and the same recursive numerical sequence, the Fibonacci numbers!
7.11.3. The Generalized “Asymmetry Principle” of Organic Nature
Using the model of “rabbit reproduction,” which is described by the tran
sitions (7.97) and (7.98), we can generalize the problem of rabbit reproduc
tion in the following manner. Let us give a nonnegative integer p≥0 and
formulate the following problem:
“Let us suppose that in the enclosed place one pair of rabbits (female and
male) is in the first day of January. This rabbit couple gives birth to a new pair
of rabbits in the first day of February and then in the first day of each follow
ing month. The newborn rabbit pair becomes mature in p months and then
gives birth to a new rabbit pair each month thereafter. The question is: how
many rabbit pairs will be in the enclosed place in one year, that is, in 12 months
from the beginning of reproduction?”
It is clear that for the case p=1 the generalized variant of the “rabbit repro
duction” problem coincides with the classical “rabbit reproduction” problem
described in Chapter 2.
Note that the case p=0 corresponds to the idealized situation when the rabbits
become mature at once after birth. One may model this case using the transition:
A→AA.
(7.99)
It is clear that the transition (7.99) reflects symmetry of “rabbit reproduc
tion” when the mature rabbit pair А turns into two identical mature rabbit pairs
АA. It is easy to show that for this case the rabbits are reproduced according to
the “Dichotomy principle,” that is, the amount of rabbits doubles each month:
1, 2, 4, 8, 16, 32, ….
Now, let us consider the case p>0. We can analyze the above “rabbit reproduc
tion” problem in greater detail, taking into consideration new conditions of “rabbit
reproduction.” It is clear that the reproduction process is described by a more com
plex system of transitions. Really, let A and B be the pairs of mature and newborn
rabbits, respectively. Then the transition (7.97) simulates a process of the monthly
appearance of the newborn pair В from each mature couple А.
Chapter 7
Algorithmic Measurement Theory
411
Let us examine the process of transformation of the newborn pair В into the
mature pair А. It is evident that during the maturation process the newborn pair
В passes via the intermediate stages corresponding to each month:
B → B1
B1 → B2
B2 → B3
(7.100)
...
Bp −1 → A
For example, for the case p=2 a process of transformation of the newborn
pair into the mature pair is described by the following system of transitions:
B → B1
(7.101)
(7.102)
B → A.
Then, taking into consideration (7.97), (7.101) and (7.102), the process
of “rabbit reproduction” for the case p=2 can be represented in Table 7.4.
Table 7.4 The process of “rabbit reproduction” for the case p=2
Date
Rabbit's pairs
A
B
B1
A + B + B1
January, 1
A
1
0
0
1
February, 1
AB
1
1
0
2
March, 1
ABB1
1
1
1
3
April, 1
ABB1 A
2
1
1
4
May, 1
ABB1 AAB
3
2
1
6
June, 1
ABABB1
ABB1 AA
4
3
2
9
July, 1
ABB1 AABABB1 ABB1 A
6
4
3
13
Note that column А gives the number of the mature pairs for each stage of
reproduction, column В gives the number of the newborn pairs, column B1 gives
the number of the newborn pairs in stage B1, column A+B+B1 gives the general
number of rabbit pairs for each stage of reproduction.
The analysis of the numerical sequences in each column
A : 1,1,1, 2, 3, 4, 6, 9,13,19,...
B : 0,1,1,1, 2, 3, 4, 6, 9,13,19,...
B1 : 0, 0,1,1,1, 2, 3, 4, 6, 9,13,19,...
A + B + B1 : 1, 2, 3, 4, 6, 9,13,19,...
demonstrates that they are subordinated to one and the same regularity: each
number of the sequence is equal to the sum of the preceding number and the
Alexey Stakhov
THE MATHEMATICS OF HARMONY
412
number distant from the latter in 2 positions. However, as we know the Fibonacci
2numbers are subordinated to this regularity!
Studying this process for the general case of р we come to the conclusion
that the Fibonacci pnumbers are a solution to the generalized variant of the
“rabbit reproduction” problem! They reflect the “Generalized Asymmetry
Principle” of “Organic Nature.”
At first appearance the above formulation of the generalized problem of
“rabbit reproduction” appears to have no real physical sense. However, we
should not hurry to such a conclusion! The article [171] is devoted to the appli
cation of the generalized Fibonacci pnumbers for simulation of biological cell
growth. The article affirms, “In kinetic analysis of cell growth, the assumption
is usually made that cell division yields two daughter cells symmetrically. The
essence of the semiconservative replication of chromosomal DNA implies com
plete identity between daughter cells. Nonetheless, in bacteria, insects, nema
todes, and plants, cell division is regularly asymmetric, with spatial and func
tional differences between the two products of division…. Mechanism of asym
metric division includes cytoplasmic and membrane localization of specific pro
teins or of messenger RNA, differential methylation of the two strands of DNA
in a chromosome, asymmetric segregation of centrioles and mitochondria, and
bipolar differences in the spindle apparatus in mitosis.” In the models of cell
growth based on the Fibonacci 2 and 3numbers are analyzed [171].
The authors of [171] made the following important conclusion: “Binary
cell division is regularly asymmetric in most species. Growth by asymmetric
binary division may be represented by the generalized Fibonacci equation ….
Our models, for the first time at the single cell level, provide rational bases for
the occurrence of Fibonacci and other recursive phyllotaxis and patterning in
biology, founded on the occurrence of regular asymmetry of binary division.”
Now we return to the Asymmetry Principle of Measurement. We can see
that the transitions (7.100) for the generalized problem of “rabbit reproduc
tion” are similar to the return of the balance to the initial position in the
BashetMendeleev problem. This means that the generalized problem of “rab
bit reproduction” is isomorphic to the Fibonacci measurement algorithm!
We can generalize this idea. As the Asymmetry Principle of Measurement is
the main idea of the algorithmic measurement theory, we can put forward the
following hypothesis. The algorithmic measurement theory described by the
very general recursive relation (7.70) and (7.71) is isomorphic to a General
Theory of Biological Populations. This means that we can use the main result
of the algorithmic measurement theory in the scientific field, which is very
far from measurement theory.
Chapter 7
Algorithmic Measurement Theory
413
7.11.4. Isomorphism between Measurement Algorithms and Positional
Number Systems
In the above we found that each measurement algorithm generates some posi
tional number system. For example, the binary measuring algorithm generates the
binary system, the optimal (n,k,0)algorithm generates all wellknown positional
number systems, including the Babylonian sexagesimal system and the decimal sys
tem. This means that the algorithmic measurement theory is isomorphic to the
theory of positional number systems and we can interpret all optimal measurement
algorithms as fundamentally new positional number systems.
It is very important to emphasize, that the Algorithmic Measurement The
ory [20, 21] as though precede a positional principle of number representa
tion and positional numeral systems. This means that the creation of a gener
al theory of positional numeral systems is reduced to a synthesis of the opti
mal measurement algorithms. This remark is very important in order to de
termine the role and place of the Algorithmic Measurement Theory in the de
velopment of mathematics. In the Introduction, we attributed the Babylo
nian positional principle of number representation to the greatest mathe
matical discovery, which preceded number theory and mathematics. The Al
gorithmic Measurement Theory infinitely enlarges a number of new, until now
unknown positional numeral systems, turning mathematics back to the peri
od of its origin. It is important to emphasize the “great practicability” of the
Algorithmic Measurement Theory, which could become the basis of a new com
puter arithmetic that is of fundamental interest to computer science.
7.12. Conclusion
1. It is well known that the “problem of measurement” played a fundamental
role in mathematics. It stimulated the development of two basic theories of
mathematics – geometry and number theory. The famous Bulgarian mathemati
cian and academician L. Iliev wrote, “During the first epoch of mathematics
development, from antiquity to the discovery of differential and integral calcu
lus, mathematics, investigating first of all the measurement problems, created
Euclidean geometry and number theory” [5]. That is why, since antiquity math
ematical measurement theory together with number theory have been consid
ered to be the fundamental mathematical theories underlying mathematics.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
414
2. Let us compare the Classical Mathematical Measurement Theory [3, 4]
with the Algorithmic Measurement Theory [20]. The creation of the classical
measurement theory is connected with internal problems that appeared in math
ematics after the discovery of Incommensurable Line Segments. This discovery
staggered the Pythagoreans and caused the first crisis in the foundations of math
ematics. The resulting Irrational Numbers became one of the fundamental
concepts of mathematics. In order to overcome the first crisis in the founda
tions of mathematics, the great mathematician Eudoxus (408 355 BC) sug
gested the Method of Exhaustion, used by him for the creation of a Theory of
Magnitudes, which preceded the Mathematical Measurement Theory complet
ed in the 19th century. The interest in mathematical measurement theory in
the 19th century increased again. The measurement theory received a new
impulse thanks to Cantor’s theory of infinite sets. The mathematical mea
surement theory was constructed on the basis of the Continuity Axioms EudoxusArchimedes’ Axiom and Cantor’s Axiom, which underlie classical
mathematical measurement theory. Thanks to Cantor’s axiom, Cantor’s idea
of Actual Infinity was introduced into mathematical measurement theory
that allowed for the proof of the Basic Measurement Equality Q=qV. This
equality for the given measurement unit V sets onetoone correspondence
between geometric magnitudes Q and real numbers q. However, a construc
tive approach to mathematics foundations, based on the Potential Infinity
concept, demands the elimination of Cantor’s axiom from the Continuity
Axioms, and puts forward a problem to prove the equality Q=qV from the
constructive point of view (Constructive Measurement Theory).
3. In contrast to the classical mathematical measurement theory, the cre
ation of the Algorithmic Measurement Theory is connected with the practical
need for new algorithms of measurement to appear in theoretical metrology,
in particular, in the theory of analogtodigital conversion [20]. It was found,
that this practical problem is deeply connected with the ancient “Problem
about the Choice of the Best Weights System.” This problem appeared in the
1202 work Liber Abaci by Fibonacci and is the first optimization problem in
measurement theory. In the Russian mathematical literature, this problem is
called the BashetMendeleev Problem [169]. The analysis of the BashetMen
deleev problem from point of view of analogtodigital conversion resulted in
the discovery of the very unusual measurement property known as the Asym
metry Principle of Measurement, which in turn underlies Algorithmic Mea
surement Theory [20, 21, 88, 90, 91]. The algorithmic measurement theory is
an absolutely new and original mathematical theory. The synthesis of opti
mal measurement algorithms is its main topic. This resulted in the discovery
Chapter 7
Algorithmic Measurement Theory
415
of the infinite number of new, unknown until now, optimal measurement algo
rithms, which have great theoretical and practical interest. These optimal
measurement algorithms contain all the classical measurement algorithms
(“binary,” “counting” and ruler” algorithms) and generate a wide class of new
original measurement algorithms based on the Fibonacci pnumbers, binomi
al coefficients, Pascal’s triangle, and so on. The main recursive relation of the
algorithmic measurement theory (7.70), which generates an infinite number
of different numerical sequences including natural numbers, binary numbers,
Fibonacci pnumbers and binomial coefficients, is of fundamental interest for
combinatorial analysis.
4. The idea of isomorphism resulted in an expansion of the applications of
the algorithmic measurement theory. There is an isomorphism between
measurement algorithms, the Fibonacci “rabbit reproduction,” and positional
number systems. This idea gives us the right to put forward a new hypothesis
that the new theory of measurement is the source of the three isomorphic
theories, namely Algorithmic Measurement Theory, New Theory of Positional
Number Systems and New Theory of Biological Populations.
5. The new theory of positional number systems returns mathematics to the
Babylonian “positional principle of number representation,” which is consid
ered to be the greatest mathematical achievement of Babylonian mathemat
ics. We can emphasize that positional number systems were never considered
in mathematics as a subject of serious mathematical research. That is why, a
new theory of positional number systems, which follows from the algorith
mic measurement theory, fills this gap, and a new theory of positional number
systems should become a fundamental part of number theory.
6. We have a new theory of biological populations based upon the isomor
phism between measurement algorithms and rabbit reproduction. In partic
ular, as formulated above, the Generalized “Asymmetry Principle” of Organic
Nature can play fundamental role here. This idea has been confirmed by con
temporary research in the field of cell division [171].
Alexey Stakhov
THE MATHEMATICS OF HARMONY
416
Chapter 8
Fibonacci Computers
8.1. A History of Computers
8.1.1. Era of Information
An interesting view of the history of computer science is given in the book
[172]. The time when people started creating instruments for hunting and man
ual labor, is usually considered to be the beginning of human civilization. The
secret of the prometheanlike “fire abduction” got lost in antiquity. However,
the subsequent history of technical progress – from the “fire abduction” up to
the discovery of nuclear energy – is the history of more and more powerful
forces of nature being subjugated by humans: taming of animals, windmills,
watermills, thermal engines, and atomic energy. During several millennia the
main problem of material culture of humanity was to increase the muscular
force of humans by various tools and machines. On the other hand, from aba
cus stones up to modern computers, the efforts to create the tools strengthen
ing human information processing mark the way for the accumulation of ideas
in the general stream of scientific and technical progress, evident in the thin
trickle of facts and museum pieces.
Already in the earliest stages of the collective work development, a per
son required some encoded signals for communication in order to synchro
nize labor operations with others. Human speech, as the predecessor of infor
mation technology and as the first transmitter of human knowledge, arose as
a result of the complication of these information signals. At first, human knowl
edge was accumulated in the form of oral stories and legends transmitted
from one generation to the next.
For the first time, the natural potential of a person accumulating and
transmitting knowledge obtained “technological support” with the creation
of Written Language. Written language became the first historical stage in
the development of information technology.
Chapter 8
Fibonacci Computers
417
We can find the proof of this stage imbedded deep in human history: such as,
cave paintings (pictographs) and carvings (petroglyphs) in the form of people and
animal images in stone, created twenty to twentyfive thousand years ago; a lunar
calendar, engraved on bone more than twenty thousand years ago. However, ac
cording to modern archaeological data, the time between the use of the first work
tools (axe, trap, etc.) and tools for representing images (in stone, bone, etc.) is rough
ly a million years. That is, nearly 99% of human history deals only with the develop
ment of material labor tools. It appears that only the last (roughly) 1% of human
history involves the development of information representation through images.
The wheel, a recognized symbol of contemporary technical progress, was sup
posedly invented in the Ancient East about 4 millennia BC, but the most an
cient written code of laws, the first monument of the written language, arose
only one thousand years later. About four millennia passed before the creation of
the first Printing Machine in the middle of 15th century proclaimed to the world
the onset of the Era of Printed Knowledge. The printing press rapidly increased
the circulation of books as a passive means of information transmission. Printing
machines for the first time created information accelerating the growth of pro
ductive forces. The creation of Computers became the next stage in the develop
ment of human civilization. These new machines for information processing ap
peared in the middle of the 20th century, when the growing load of information
problems became one of the most noticeable factors, limiting economic growth
in industrially developed countries.
Thus, during the history of human civilization before the 20th century, ma
terial objects were the main subject of labor, and the economic power of
countries was measured by material resources. At the end of the 20th century
for the first time in the history of humanity, information became the main sub
ject of labor in industrially developed countries. The tendency for a steady in
crease in the information sphere in comparison to the material sphere is the most
noticeable symptom of the “information era” approach.
8.1.2. The Basic Milestones in Computer Progress
The history of computers, that is, the machines created for automation of
information processing, is considered one of the least developed themes in the
history of science. In the present book we will use a classification of periods of
the computer development suggested in the book [173]. According to [173], the
history of computers is divided into two main periods:
1. From abacus up to electronic computers
2. Electronic computers onward
Alexey Stakhov
THE MATHEMATICS OF HARMONY
418
8.1.2.1. Abacus
The necessity for calculation arose in the early stages of human civilization:
notches on bones and stones representing calculations were found beginning
in the Paleolithic and Neolithic periods. The abacus was probably the first
calculating device. The word “abacus” originates from the Greek word abax,
which in turn comes from the Hebrew word avak (dust). The history of science
testifies to the fact that many different cultures used the abacus – Babylon,
China, India, Egypt, Arabian countries, Japan, Pakistan, and others.
In medieval times, the word “abacus” was used as a synonym for
mathematics. This was the meaning contained in the title of the mathematical
book, Liber Abaci, written in 1202 by the famous Italian mathematician
Leonardo from Pisa (Fibonacci). His book could be considered the original math
ematical encyclopaedia of the Middle Ages.
It is important to emphasize that in this early period of scientific
development, mathematical history is inseparably linked to the history of
primitive computers (abaci), because the problem of counting and calculation
hastened the development of many important mathematical concepts. Note
that at this stage, the Babylonian mathematicians discovered the positional
principle of number representation underlying modern computers. The
majority of the historians of mathematics emphasize that the creation of the
first primitive abacus played a large role in the development of mathematics,
and resulted in the development of a concept of natural number – one of our
fundamental mathematical concepts.
8.1.2.2. Mechanical Calculation Machines
Already philosophers in the Middle Ages advanced the problem of replacing
human brain functions with various mechanisms. Several scientists were interest
ed in the idea of the construction of “thinking” gear, and this idea stimulated the
designs of the first mechanical calculating machines. In 1623, English professor of
mathematics Schickard, a friend of Kepler, made such an attempt. In Kepler’s ar
chive we find Schickard’s letter addressed to him. Schickard informed Kepler about
a calculation machine he had constructed. Schickard’s machine was known, ap
parently, by a very small group of people. Because of this limited exposure, for a
long time it was thought that the first mechanical calculator was invented in 1642
by the famous French mathematician and physicist Pascal. About 50 copies of
Pascal’s calculator were purportedly produced. Some copies of Pascal’s calculator
are still available today. Five copies are in the Parisian Museum and one is in the
Dresden physical and mathematical salon, which this author personally saw in
Chapter 8
Fibonacci Computers
419
1988. In addition, the German mathematician Gottfried Leibniz, Wurttemberg’s
pastor Gan, and the famous Russian mathematician and mechanician Chebyshev
were inventors of original mechanical calculating machines.
Production of calculating machines for the first time was started by Karl
Thomas, the founder and chief of two Parisian insurance companies (“Fenix”
and “Soleil”). In 1820, he built the calculation machine called Calculator. In 1821,
15 calculators were made in Thomas’ workshop. Later their production was in
creased to 100 per year. These calculators had been produced during 100 years
and had (certainly for that time) very impressive technical characteristics – two
8digit numbers could be multiplied in 15 seconds, and a 16digit number could
be divided by the 8digit number in 25 seconds.
A further improvement of calculators is connected with the name of Russian
engineer Odner. Beginning in the 1890s, the triumphal procession of Odner’s
calculators began with the serial production of 500 calculators per year in St.
Petersburg, Russia. In the 20th century, Odner’s calculators were produced un
der a variety of names in different countries. In the first quarter of the 20th cen
tury, Odner’s calculators were the main mathematical machines used in many
fields of human activity. In 1914, in Russia alone more than twentytwo thou
sand Odnercalculators were produced.
8.1.2.3. Babbage’s Analytical Machine
Charles Babbage was the mathematician, mechanical engineer and com
puter scientist who originated the idea of a Programmable Computer. Babbage’s
analytical machine was the best of the mechanical calculation machines. For the
first time this machine embodies the idea of a completely automatic computer
with program control. This idea was a revolutionary discovery in computer sci
ence. It started to be used only in the middle of the 20th century in the design of
the first electronic computers. The first copy of
his analytical machine was made by Babbage in
September 1834.
From 1842 to 1848, Babbage paid significant
attention to the design of his analytical machine.
According to his idea, the machine should include
three main blocks:
1. A device where digital information could
be stored on sprocket registers. In modern
computers a similar block is named the Stor
age Device or Memory. Babbage named this
part of the machine the Storehouse.
Charles Babbage (1791 –1871)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
420
2. A device where various operations with numbers, taken from the store
house, can be carried out. Today this part of the machine is called the Arith
metical Device. Babbage named this part of the machine the Factory.
3. A device for control of the operating sequence. Babbage didn’t have a spe
cial name for this part of the machine. Today a similar computer block is
called the Control Unit.
In addition, his analytical machine had both Input and Output Devices.
Babbage was the creator of the greatest invention in computer history,
the Programmable Computer. Babbage’s main idea was only embodied in
modern computers. A simple list of problems that was advanced by Babbage
in his analytical machine is astonishing for its depth and foresight regarding
the progress of modern computers.
8.1.2.4. Hollerith’s Tabulators
The development of the theory of electricity and the theory of weak electri
cal currents naturally resulted in the idea of using these currents in calculation
devices. At first, the electric energy in calculation machines was used only as a
motion force for the actuation of the calculator mechanism in place of the hand.
Such machines were called Electromechanical Machines.
At the end of the 19th century, the need for population census processing arose.
In 1888, an employee of the U.S. Census Bureau, Herman Hollerith, designed a
calculation machine that was used for processing the population census. Hollerith
named his machine the Tabulator. His tabulator was intended to automatically
process punch cards for the population census. In his machine Hollerith applied
the benefits of weak current engineering and his machine was based on the electro
mechanical principle. The same electromechanical principle was used for improve
ment of the technical parameters of calculators. Thus, at the beginning of the 20th
century, digital computer engineering developed in two directions. The first di
rection was the small Calculators intended for the mechanization of the elementa
ry arithmetical operations. The second direction was that of Tabulators intended
for the processing of statistical information.
8.1.2.5. Electromechanical Computers by Zuse, Aiken and Stibitz
In the 1930s and 1940s the development of tabulators, as well as, the achieve
ments in the field of the electromechanical relay applications (for example, in
automatic telephone stations) resulted in the development of computer machines
similar to Babbage’s machine by their structure. The first universal electrome
chanical computers with programmed control were designed in Germany and
the U.S. in the early 1940s. German engineer Konrad Zuse, along with American
Chapter 8
Fibonacci Computers
421
scientists Howard Aiken from Harvard University and George Robert Stibitz
from Bell Labs, played significant roles in these projects.
Zuse developed several models of electromechanical computers. The first
model Z1 was constructed over a period of two years (19361938). It was based
completely on mechanical elements, rendering it somewhat unsatisfactory. The
next models Z2, Z3, Z4 used electromagnetic relays as their basis. An elabora
tion of the Z3 machine was completed in 1941. This machine was the first uni
versal computer with programmed control. The computer performed eight com
mands including four arithmetical operations, multiplying by negative numbers,
and calculating square roots. All calculations were fulfilled in the binary system
using a floating point.
Aiken’s universal digital computer used standard punch cards produced
by IBM (USA). In August, 1944 the project was completed, and the machine
called MARC1 was installed at Harvard University, where the machine was
used thereafter for over 15 years. If we compare the machine MARC1 with
the machine Z3, we should note that Zuse’s machine exceeded Aiken’s
machine from the point of view of its circuit and structural solutions.
Also the works of American scientist Stibitz played an essential role in
the creation of the first computers, which were intended for the fulfilment of
complicated scientific and technical calculations. In 1938, working for Bell
Laboratory, Stibitz developed the computer “Bell1” capable of operating
with complex numbers. In 1947, Stibitz developed a highpower universal
computer with programmed control (the machine “Bell5”) based on elec
tromagnetic relays. At the same time, the computing laboratory of Harvard
University developed the large programcontrolled computer MARC2 which
is also based on electromagnetic relays.
8.1.2.6. ENIAC
One of the first attempts to use electronic elements in computers was
launched in the U.S. in 19391941 at the State College of Iowa (now the Iowa
State University). Physics professor John Atanasov was the principal developer
of this project. Unfortunately, the project was not completed, but in 1973 the
U.S. Federal Court confirmed Atanasov’s priority in the design of the first elec
tronic computer project.
Work on the creation of the electronic computer ENIAC was started in 1942,
at Pennsylvania University under John Mauchly and John Eckert’s leadership.
This works was completed at the end of 1945, and in February 1946 the ma
chine’s first public demonstration was held. The important role of ENIAC in the
development of computer science is clear from the fact that it was the first ma
Alexey Stakhov
THE MATHEMATICS OF HARMONY
422
chine in which electronic elements were used for the realization of arithmetical
and logical operations, and also the storage of information. The use of new elec
tronic technology in this machine allowed for an increase of computer operation
speed by approximately 1000 times greater in comparison with electromechan
ical computers.
8.1.2.7. John von Neumann’s Principles
The ENIAC confirmed in practice the high efficiency of electronic technolo
gy in computers. The problem of maximal real
ization of the large advantages of electronic tech
nology arose for computer designers. It was nec
essary to analyze the strengths and weaknesses of
the ENIAC project and to give appropriate rec
ommendations. A brilliant solution to this prob
lem emerged in the famous report Preliminary Dis
cussion on Logic Designing the Electronic Com
puting Device (1946). This report, written by
famed mathematician John von Neumann and his
colleagues from Princeton University, Goldstein
and Berks, presented the project of developing the
John von Neumann
(19031957)
new electronic computer.
The essence of the primary recommendations
of the report consisted of the following:
1. The machines based on electronic elements should work in Binary System
instead of the decimal system.
2. The program should be stored in the machine block called the Storage
Device, which should have sufficient capacity and appropriate speeds for ac
cess and entry of a variety of program commands.
3. Programs, as well as numbers, with which the machine operates, should be
represented in Binary Code. Thus, Commands and Numbers Should Have
One and the Same Form of Representation. This meant that all programs
and all intermediate outcomes of calculations, constants and other numbers
should be stored in one and the same storage device.
4. Difficulties of physical realization of the storage device, speed of which
should correspond to the speed of logical elements, demanded a Hierarchical
Organization of Memory.
5. The arithmetical device of the machine should be constructed on the basis
of the Logical Summation Element; it is not recommended to design special
devices for the realization of other arithmetical operations.
Chapter 8
Fibonacci Computers
423
6. The machine should use the Parallel Principle of Organization of Com
puting Processes, that is, the operations on binary words should be performed
on all digits simultaneously.
John von Neumann’s Principles, in general, were some of the most important
contributions to the development of universal electronic computers. The further
development of electronic computers was based upon these principles. The changes
concerned only the technology of the element base. Depending on the electronic
base used, electronic computers started developing in the following three directions:
1. Computers based on the electronic lamps.
2. Computers based on the discrete semiconducting and magnetic elements.
3. Computers based on the integrated schemes.
8.1.2.8. The Phenomenon of Personal Calculators
The capacity of the present book, which is not a specialty textbook on the
history of the computer, does not allow us to refer to many achievements of mod
ern computers. We can only discuss one phenomenon of modern computer his
tory – that of personal computers (PC), which were the first mass tools for the
active formalization of professional knowledge. Varied influence of the PC on
progress of industrial society is comparable to book printing, which began the
information era. Continuing this analogy: while a book was and remains the tool
for mass duplication and passive storage of knowledge, the PC is the first mass
produced tool for direct transformation of professional knowledge into an active
industrial product. The discovery of the PC phenomenon in the U.S. was con
nected with the name Steve Jobs, founder of Apple Computer.
In 1980, Steve Jobs defined this type of computer as the tool for increasing the
natural potentialities of human intellect. The PC, more than any other tool for infor
mation processing, brings forward the information era. According to the opinion of
many competent scientists, we can now identify a few of the symptoms, which testi
fy to the beginning of the transition within the industrially developed countries to a
qualitatively new stage of technological development – the information era:
1. Information becomes the main subject of labor in public production of the
industrially developed countries.
2. The time of doubling accumulated scientific knowledge is something like
23 years.
3. The material expenses for storage, transmission and processing of infor
mation begin to exceed similar costs for energy expenses.
4. Humanity for the first time in its history becomes the really observed astro
nomical “cosmic factor,” as the level of Earth radio emissions in the separate
parts of the radio diapason approaches the level of the Sun’s radio emissions.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
424
8.2. Basic Stages in the History of Numeral Systems
The creation of the first numeral systems is attributed to the period of the
mathematics origin, when the necessity to count things, and measure time, land
and product quantities, resulted in the development of the basic principles of
the arithmetic of natural numbers. In the history of numeral systems we can
identify several stages: the Initial Stage of Counting, NonPositional Numeral
Systems, Alphabetic Notations, and Positional Numeral Systems. Initially people
used body parts, fingers, sticks, knots, etc. for representation of counted sets.
Article [2] emphasizes that “despite the extreme primitiveness of this way of
number representation, it played an exclusive role in the development of the
number concept.” This statement confirms that numeral systems played an im
portant role in the formation of the concept of natural numbers – one of mathe
matics’ most fundamental concepts.
8.2.1. Babylonian Sexagecimal Numeral System
At an early stage in the history of mathematics, one of the greatest mathemat
ical discoveries was made. This was the Positional Principle of number representa
tion. “The Babylonian sexagecimal numeral system that arose in approximately
2000 B.C. was the first numeral system based on the positional principle” [2]. There
were a number of competing hypotheses explaining its origin. M. Cantor original
ly assumed that Sumerians (the primary population of the Euphrates valley) con
sidered that one year consisted of 360 days and therefore that base 60 had an as
tronomical origin. According to Kevitch’s hypothesis, in the Euphrates valley two
persons met. The first person knew the decimal system, and the other one knew
the base 6 system (Kevitch explains the occurrence of such base by finger count
ing, in which the compressed hand meant the number 6). After the two systems
were merged the compromise base of 60=6×10 arose.
We should note that Cantor’s and Kevitch’s hypotheses concerned the ques
tion of the origin of base 60, but they do not explain the origin of the positional
principle of number representation. Neugebauer’s hypothesis presented in the
book [174] gives an answer to this last question. According to his hypothesis,
the positional principle has a measurement origin.
Neugebauer asserts [174] that “the basic stages of the origin of the positional
principle in Babylon were the following: (1) determination of a quantitative ratio
between two independent existing systems of measures and (2) omitting the names
of numerals in the writing.” According to his opinion, these stages of the origin of
the positional numeral systems had a common foundation. He emphasizes that
Chapter 8
Fibonacci Computers
425
“the positional sexagecimal number system was the natural outcome of a long his
torical development, which does not differ from similar processes in other cultures.”
However, we can explain the choice of the number 60 as the base of the Baby
lonian system from the point of view of the Universal Harmony concept. As is well
known, the dodecahedron was the main geometric figure expressing the Universal
Harmony in ancient science. As is shown in Chapter 3, the numerical parameters
of the dodecahedron (12, 30, 60) were connected to the basic cycles of the Solar
system (Jupiter’s 12year cycle, Saturn’s 30year cycle and the Solar system’s 60
year main cycle). The ancient scientists chose the numbers 12, 30, 60 and derived
from them the number 360=12×30 as the “main numbers” of the calendar system
and systems of time and angular values measurement. Taking into consideration
the deep connection between the Babylonian and Egyptian cultures, we may ad
vance the hypothesis that the base 60 of the Babylonian number system was cho
sen by the Babylonians from astronomical considerations. This hypothesis coin
cides with Cantor’s hypothesis about the astronomical origin of base 60.
8.2.2. Alphabetic and Roman Systems of Numeration
Many people used the socalled alphabetic system of numeration, where nu
merical values were attributed to the letters of the alphabet. For instance, ancient
Greek, Hebrew, Sanskrit, Slavic and Arabic alphabets employed this idea.
The Roman system of numeration is widely known. It uses special numerals
for the representation of the “nodal” numbers (I, V, X, L, C, D, M). The greatest
shortcoming of Roman numeration was the fact that it was not adapted to per
form arithmetical operations in written form.
8.2.3. Mayan System
The origin of positional numeral systems is considered to be one of the main
events in the history of material culture. Many people took part in its creation.
In the 6th century AD a similar numeral system arose within the Maya people.
It is widely accepted that the number 20 is the base of the Mayan number sys
tem and, therefore, may have a “finger/toe” origin (10+10=20). It is well known
that the number system with base 20 used the following set of the “nodal” num
bers for number representation: {1, 201, 202, 203}. However, the Mayan number
system used another set of “nodal” numbers: {1, 20, 360, 20×360, 202×360,…}. This
means that the next “nodal” number, which follows after 20, is equal to 360 (in
stead of 202=400) and all subsequent “nodal” numbers are derivable from the
numbers 20 and 360. As is emphasized in the article [2], this somewhat unusual
numbering system can be explained by the fact that “the Maya divided one year
Alexey Stakhov
THE MATHEMATICS OF HARMONY
426
into 18 months, with 20 days in every month, plus five more days.” Thus, like
the base of the Babylonian system, the “nodal” numbers of Mayan notation have
an astronomical origin. It is essential to emphasize, that the structure of the Maya
calendar year (360+5=18×20+5) is similar to the structure of the Egyptian cal
endar year (360+5=12×30+5).
8.2.4. Decimal System
We use a decimal system for daily calculations. In mathematics this numeral
system is called the HinduArabic or Arabic numeral system. This number system
was invented by Hindu mathematicians in approximately the 6th8th centuries
AD and it is the predecessor of the traditional decimal system. This number sys
tem uses the number 10 as the base and requires 10 different numerals, the digits
0, 1, 2, 3, 4, 5, 6, 7, 8, 9 for the representation of numbers. From India, the decimal
system penetrated into the Arabic world. Arabic mathematicians extended the
system to decimal fractions. In 9th century Muhammad ibn Musa alKhwarizmi
wrote an important work about the decimal system. Thanks to a translation of al
Khwarizmi’s work, the decimal system began to be used in Europe.
There are different opinions concerning the choice of the number 10 as the
base of the decimal system; the most widespread opinion is that the base of 10
has a “finger” origin. However, it is necessary to remind oneself that the number
10 always had special significance in ancient science. Pythagoreans considered
this number to be the “Sacred Number” and named it the Tetractys. The number
10=1+2+3+4 was considered by Pythagoreans to be one of the greatest values,
being “a symbol of the Universe,” because it comprised four “basic elements”:
the One or Monad symbolizes spirit, out of which all the visible world appears;
the Two or Dyad (2=1+1), symbolizes the material atom; the Three or Triad
(3=2+1), symbolizes the living world. The Four or Tetrad (4=3+1) connected
the living world with the Monad and consequently symbolizes the Whole, that
is, the Visible and Invisible World together. As the Tetractys 10=1+2+3+4, which
meant that the Tetractys, 10, expressed Everything. Thus, the hypothesis about
the “harmonic” origin of base 10 is as tenable as the hypothesis of the “finger”
origin of the decimal system.
8.2.5. Binary System
The binary system uses two symbols, 1 and 0, for number representation.
The number 2 is its base or radix. In connection with the development of com
puter technology, the binary system was introduced into modern science. Dis
cussing the history of the binary system, we should note that the ancient Indian
Chapter 8
Fibonacci Computers
427
writer and mathematician Pingala pre
sented in 2nd century AD the first
known description of a binary system,
more than 1500 years before its discov
ery by German mathematician Got
tfried Leibniz in 1695. Pingala’s work
also contains elements of combinatorics
(Pascal triangle) and the basic ideas of
Fibonacci numbers.
The modern binary system was ful
ly documented by Gottfried Leibniz in
17th century in his article Explication
de l’Arithmйtique Binaire. Leibniz’s sys
tem used 0 and 1 in a manner similar to
the modern binary system. Gottfried
Wilhelm von Leibniz (1646 1716) was
Gottfried Wilhelm von Leibniz
a German polymath who wrote mostly
(16461716)
in French and Latin. He invented calcu
lus independently of Newton. Leibniz also made major contributions to physics
and technology, and anticipated notions that surfaced much later in biology, med
icine, geology, probability theory, psychology, and information science. His con
tributions to this vast array of subjects are scattered in journals and in numerous
letters and unpublished manuscripts. To date, there is no complete edition of Leib
niz’s writings, and a full list of his accomplishments is not yet known.
The binary arithmetic developed by Leibnitz is much similar to arithmetic
in other numeral systems. Summation, subtraction, multiplication, and division
can be performed on binary numerals. All arithmetical operations in the binary
system are reduced to binary summation. The summation of two singledigit
binary numbers is relatively simple:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (carry of 1)
In 1854, British mathematician George Boole published an important paper
detailing a system of logic that became known as Boolean Algebra. His logic
system became the instrumental tool for the development of the binary system,
particularly in its implementation in electronic circuits.
In 1937, Claude Shannon presented his master’s thesis at MIT that initiated
the implementation of Boolean algebra and binary arithmetic using electronic
Alexey Stakhov
THE MATHEMATICS OF HARMONY
428
relays and switches for the first time in history. Entitled A Symbolic Analysis of
Relay and Switching Circuits, Shannon’s thesis essentially developed the practi
cal digital circuit design.
In 1946, John von Neumann, Goldstein and Berks in their Preliminary
Discussion on Logic Designing the Electronic Computing Device gave a decisive
preference for the use of the binary system in electronic computers.
8.2.6. Exotic Number Systems
However, we should note that the binary system suffers from some serious
disadvantages [175, 176]:
1. The problem of a number sign. In the binary system we can represent in
the “direct” code only positive numbers. That is why, for the representation
of negative numbers we need to use special codes additional and inverse
codes. This complicates the arithmetical structures of computers and influ
ences the internal performance of a computer.
2. The problem of error detection. As is well known, all computer devices are
subjected to severe external and internal influences that are a cause of vari
ous errors, which can appear in the information transmission and process
ing. Unfortunately, such errors cannot be detected because all binary code
combinations are permissible.
3. The problem of long carryover. If, for example, we sum the two binary
numbers 011111111+000000001=100000000, we should take into consider
ation the “long carryover” from the lowest digit to the highest digit appear
ing in such a summation. This “long carryover” fundamentally influences
the internal performance of computers.
The intention to overcome these disadvantages of the binary system led to the
appearance of different numeral systems with “exotic” titles and properties: Sys
tem for Residual Classes, Ternary Symmetrical Numeral System, Numeral System
with Complex Radix, Negapositional, Factorial, Binomial Numeral Systems [175,
176], etc. All of them had certain advantages in comparison with the binary sys
tem, and were directed at the improvement of computer characteristics; some of
them became the basis for the creation of new computer projects (e.g. the ternary
computer “Setun,” the computer based on system for residual classes, and so on).
However, there is also another interesting aspect to this problem. Four mil
lennia after the invention by Babylonians of the positional principle of number
representation, we observe some kind of “Renaissance” in the field of numeral
systems. Thanks to the efforts of the experts in computers and mathematics, it is
as if we once again have returned to the origin of computation, when the numeral
Chapter 8
Fibonacci Computers
429
systems defined the subjectmatter and were the heart of mathematics (Baby
lon, Ancient Egypt, India, and China).
The main purpose of chapters 8, 9 and 10 is to demonstrate that new meth
ods of number representation based on the golden mean and Fibonacci numbers
can become the source of a new and unusual computer arithmetic and from this
will follow new and exciting computer projects.
8.3. Fibonacci pCodes
8.3.1. Zeckendorf Sums
Many numbertheorists know about Zeckendorf sums [16], however, few know
about the man for whom these sums are named. The Fibonacci Association con
siders Edouard Zeckendorf to be one of the famous Fibonacci mathematicians of
the 20th century although he did not have mathematical education. Zeckendorf
was a medical doctor and was licensed for dental surgery. Mathematics was Zeck
endorf’s hobby. Through the years, he published several mathematical articles.
The most important article, on Zeckendorf sums, was published in 1939. In it,
Zeckendorf proved that each positive integer can be represented as the unique
sum of nonadjacent Fibonacci numbers, as exemplified below:
38=34+3+1; 39=34+5; 40=34+5+1; 41=34+5+2; 42=34+8.
The numerous articles published in The Fibonacci Quarterly discussed Zeck
endorf sums and their generalizations.
Let us write Zeckendorf representation in the following form:
N=anFn+an1Fn1+…+aiFi+…+a1F1,
(8.1)
where ai ∈{0,1} is the binary numeral of the ith digit of the representation (8.1);
n is the number of digits of the representation (8.1); Fi is a Fibonacci number
given by the recursive relation
Fn= Fn1+Fn+2
at the seeds
(8.2)
(8.3)
F1= F2=1.
It is important to emphasize that the Fibonacci numbers Fi (i=1,2,3,…,n) are
used as the digit weights in Zekendorf’s representation (8.1).
The binary representation of Zeckendorf sum (8.1) has the following form:
(8.4)
N = an an1 ... ai ... a1,
where ai ∈{0,1} is a binary numeral of the ith digit of the binary representation
Alexey Stakhov
THE MATHEMATICS OF HARMONY
430
(8.4). Thus, Zeckendorf sum (8.1) can be considered as the binary representa
tion (8.4) with “Fibonacci weights.”
It is clear that the minimal number Nmin that can be represented by using
Zeckendorf sum (8.1) is equal to 0 and has the following binary representation:
N min = 0 = 00
...0 .
N
(8.5)
n
The maximal number Nmax that can be represented by using Zeckendorf sum
(8.1) has the following binary representation:
N max = 11
...1.
N
(8.6)
n
The binary representation (8.6) is an abridged representation of the
following sum:
Nmax=Fn+Fn1+…+Fi+…+F1
Using the formula (2.20), we can write:
(8.7)
(8.8)
Nmax=Fn+21.
This means that we have proved the following theorem.
Theorem 8.1. Using the ndigit Zeckendorf sum (8.1), we can represent
Fn+2 integers in the range from 0 to Fn+21.
8.3.2. Definition of the Fibonacci pCode
In Chapter 7 we developed the socalled Algorithmic Measurement Theory
[20, 21] that can be a source of new ideas in numeral systems field. Fibonacci
codes and Fibonacci arithmetic, which are a wide generalization of the binary
system, are amongst the most interesting applications of the algorithmic mea
surement theory.
The Fibonacci measurement algorithms, which were examined in Chapter
7, are isomorphic to the following sum:
(8.9)
N=anFp(n)+an1Fp(n1)+…+aiFp(i)+…+a1Fp(1),
where N is a natural number, ai ∈{0,1} is a binary numeral of the ith digit of the
code (8.9); n is the digit number of the code (8.9); the Fibonacci number Fp(i) is
the ith digit weight calculated in accordance with the recursive relation (4.18)
at the seeds (4.19).
The positional representation of the natural number N in the form (8.9) is
called a Fibonacci pcode [20]. The abridged representation of the Fibonacci
pcode (8.9) has the same form (8.4) as the abridged representation of Zecken
dorf sum (8.1).
Note that the notion of a Fibonacci pcode (8.9) includes an infinite number
of different positional “binary” representations of natural numbers because ev
Chapter 8
Fibonacci Computers
431
ery p produces its own Fibonacci pcode (p=0,1,2,3,…). In particular, for the case
p=0 the Fibonacci pcode (8.9) is reduced to the classical binary code:
(8.10)
N=an2n1+an12n2+…+ai2i1+…+a120 .
For the case p=1 the Fibonacci pcode (8.9) is reduced to Zeckendorf
sum (8.1).
Now, let us consider the partial case p=∞. For this case each Fibonacci p
number is equal to 1, that is, for any integer i we have:
Fp(i)=1.
Then the sum (8.9) takes the form of the “unitary code”:
N = 1+ 1 + ... + 1.
(8.11)
Thus, the Fibonacci pcode given by (8.9) is a very wide generalization of
the binary system (8.10) and Zeckendorf sum (8.1) that are partial cases of the
Fibonacci pcode (8.9) for the cases p=0 and p=1, respectively. On the other
hand, the Fibonacci pcode (8.9) includes the socalled “unitary code” (8.11) as
another extreme case for p=∞.
N
8.3.3. The Range of Number Representation in the Fibonacci pCode
Consider the set of ndigit binary words. The number of them is equal to 2n.
For the classical binary code (8.10) (p=0) the mapping of the set of ndigit bina
ry words onto the set of natural numbers has the following peculiarities:
(a) Uniqueness of mapping. This means that for the infinite n there are one
toone correspondences between natural numbers and sums (8.10), that is,
every integer N has only one representation in the form (8.10).
(b) For a given n with the help of the binary code (8.10) we can represent all
integers in the range from 0 to 2n 1, that is, the range of number representa
tion is equal to 2n.
(c) The minimal number 0 and the maximal number 2n1 have the following
binary representations in the binary code (8.10), respectively:
N min = 0 = 00
...0
N
n
N max = 2n − 1 = 11
...1.
N
n
For the Fibonacci pcode (8.9) the mapping of the ndigit binary words onto
natural numbers has distinct peculiarities for the case p>0.
Let n=5. Then, for the cases p=1 and p=2 the mappings of the 5digit Fi
bonacci pcode (8.9) onto the natural numbers have the forms, represented in
Tables 8.1 and 8.2, respectively.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
432
Table 8.1. Mapping the Fibonacci 1code
onto natural numbers
An analysis of Tables 8.1
and 8.2 leads to the following
CC 5 3 2 1 1 N CC 5 3 2 1 1 N
peculiarities of binary represen
A0 0 0 0 0 0 0 A16 1 0 0 0 0 5
tations of natural numbers in
A1 0 0 0 0 1 1 A17 1 0 0 0 1 6
Fibonacci pcodes (8.9). By us
A2 0 0 0 1 0 1 A18 1 0 0 1 0 6
ing the 5digit Fibonacci 1code
A3 0 0 0 1 1 2 A19 1 0 0 1 1 7
(Table 8.1), we can represent 13
A4 0 0 1 0 0 2 A20 1 0 1 0 0 7
integers in the range from 0 to
A5 0 0 1 0 1 3 A21 1 0 1 0 1 8
12, inclusively. Note that the
A6 0 0 1 1 0 3 A22 1 0 1 1 0 8
number 13 is the Fibonacci 1
A7 0 0 1 1 1 4 A23 1 0 1 1 1 9
number with index 7, that is,
A8 0 1 0 0 0 3 A24 1 1 0 0 0 8
F1(7)= F7=13. We can see from
A9 0 1 0 0 1 4 A25 1 1 0 0 1 9
Table 8.2 that by using the 5
A10 0 1 0 1 0 4 A26 1 1 0 1 0 9
digit Fibonacci 2code (Table
A11 0 1 0 1 1 5 A27 1 1 0 1 1 10
8.2) we can represent 9 integers
A12 0 1 1 0 0 5 A28 1 1 1 0 0 10
in the range from 0 to 8, inclu
A13 0 1 1 0 1 6 A29 1 1 1 0 1 11
sively. Here, the number 9 is the
A14 0 1 1 1 0 6 A30 1 1 1 1 0 11
Fibonacci 2number with index
A15 0 1 1 1 1 7 A31 1 1 1 1 1 12
8, that is, F2(8)=9. The results
of this consideration are partial cases of the following theorem.
Theorem 8.2. For the giv
Table 8.2. Mapping the Fibonacci 2code
en integers n>0 and p>0, with
onto natural numbers
the help of the ndigit Fibonac
CC 3 2 1 1 1 N CC 3 2 1 1 1 N
ci pcode, we can represent
A0 0 0 0 0 0 0 A16 1 0 0 0 0 3
F p (n+p+1) integers in the
A1 0 0 0 0 1 1 A17 1 0 0 0 1 4
range from 0 to Fp(n+p+1)1,
A2 0 0 0 1 0 1 A18 1 0 0 1 0 4
inclusively.
A3 0 0 0 1 1 2 A19 1 0 0 1 1 5
Proof. It is clear that the
A4 0 0 1 0 0 1 A20 1 0 1 0 0 4
minimal number N min can be
A5 0 0 1 0 1 2 A21 1 0 1 0 1 5
represented in the Fibonacci p
A6 0 0 1 1 0 2 A22 1 0 1 1 0 5
code (8.9) as follows:
A7 0 0 1 1 1 3 A23 1 0 1 1 1 6
N min = 0 = 00
...0 .
N
n
(8.12)
The binary representa
tion of the maximal number
Nmax has the following form:
N max = 11
...1.
N
n
(8.13)
A8
A9
A10
A11
A12
A13
A14
A15
0 1 0
0 1 0
0 1 0
0 1 0
0 1 1
0 1 1
0 1 1
0 1 1
0 0
2
0
1
1
0
0
1
1
3
3
4
3
4
4
5
1
0
1
0
1
0
1
A24
A25
A26
A27
A28
A29
A30
A31
1 1 0 0 0
5
1
1
1
1
1
1
1
6
6
7
6
7
7
8
1
1
1
1
1
1
1
0
0
0
1
1
1
1
0
1
1
0
0
1
1
1
0
1
0
1
0
1
Chapter 8
Fibonacci Computers
433
The binary combination (8.13) is an abridged representation of the fol
lowing sum:
Nmax=Fp(n)+ Fp(n1)+…+Fp(i)+…+Fp(1).
Using the formula (4.39), we can write:
(8.14)
Nmax=Fp(n+p+1)1.
(8.15)
The theorem is proved.
Note that for the case p=0, F0(n+1)=2n and Theorem 8.2 is reduced to the
wellknown theorem about the range of number representation for the classical
binary system. This range is equal to 2n for the ndigit binary code (8.10).
8.3.4. Multiplicity of Number Representation
The Multiplicity of number representation in the form (8.9) is the next
peculiarity of the Fibonacci p
Table 8.3. Mapping natural numbers to binary
code (8.9) for the case p>0. By
combinations in the Fibonacci 1 and 2 codes
accepting the minimal number
p =1
p=2
0 given by (8.12) and the max
0 = { A0 }
imal number given by (8.13),
1 = { A1 , A2 }
the remaining integers from
2 = { A3 , A4 }
0 = { A0 }
the range [0, F p(n+p+1)1]
1 = { A1 , A2 , A4 }
3 = { A5 , A6 , A8 }
have more than one represen
tation in the form (8.9). This
2 = { A3 , A5 , A6 , A8 }
4 = { A7 , A9 , A10 }
means that all integers in the
3 = {A7 , A9 , A10 , A12 , A16 }
5 = { A11 , A12 , A16 }
range [1, Fp(n+p+1)2] have
6 = { A13 , A14 , A17 , A18 } 4 = {A11 , A13 , A14 , A17 , A18 , A20 }
multiple representations in the
7 = { A15 , A19 , A20 }
5 = { A15 , A19 , A21 , A22 , A24 }
Fibonacci pcode (8.9) for the
8 = { A21 , A22 , A24 }
6 = {A23 , A25 , A26 , A28 }
case p>0.
9 = { A23 , A25 , A26 }
7 = { A27 , A29 , A30 }
Now, let us examine the
10 = { A27 , A28 }
8 = { A31 }
mapping of integers onto the 5
digit binary code combinations
11 = { A29 , A30 }
A in accordance with Table 8.1
12 = { A31 }
(p=1) and Table 8.2 (p=2).
This mapping is represented in Table 8.3.
Note that for the arbitrary p the minimal and the maximal numbers have
only binary representations in the ndigit Fibonacci pcode:
0 = 0 0 ... 0 (n digits)
and
Fp(n+p) 1 = 1 1 ... 1 (n digits).
Alexey Stakhov
THE MATHEMATICS OF HARMONY
434
8.4. Minimal Form and Redundancy of the Fibonacci pCode
8.4.1. Convolution and Devolution
The different binary representations of one and the same integer N in the
Fibonacci pcode (8.9) for the case p>0 may be obtained from one another by
means of the peculiar code transformations called Convolution and Devolution
of the binary digits. These code transformations are carried out within the scope
of one binary combination and follow from the basic recursive relation (4.18)
that connects adjacent digit weights of the Fibonacci pcode (8.9). The idea of
such code transformations consists in the following.
Let us consider the abridged representation (8.4) of integer N in the Fibonacci
pcode (8.9). Suppose that the binary numerals of the lth, (l1)th and (lp1)
th digits in (8.4) are equal to 0, 1, 1, respectively, that is,
N=an an1 … al+1 01 al2… alp 1 alp2… a1.
(8.16)
This binary representation (8.16) may be transformed into another binary
representation of the same number N, if, in accordance with the recursive rela
tion (4.18), we replace the binary numerals 1 in the (l1)th and (lp1)th digits
with the binary numerals 0 and the binary numeral 0 in the lth digit with the
binary numeral 1, that is,
an ... al +1 0 1 al −2 ... al − p 1 al − p −2 ... a1

.
N =
...
a
a
a
a
a
a
...
1
0
...
0
l − p −2
1
l +1
l −2
l−p
 n
(8.17)
Such transformation of the binary representation (8.17) is named Convolu
tion of the (l1)th and (lp1)th digits into the lth digit.
The initial binary representation (8.17) can be restored if we fulfill the fol
lowing code transformation over the binary representation of the number N:
an ... al +1 1 0 al −2 ... al − p 0 al − p −2 ... a1

.
N =
(8.18)
an ... al +1 0 1 al −2 ... al − p 1 al − p −2 ... a1
Such transformation of the code representation is named Devolution of the
lth digit into the (l1)th and (lp1)th digits.
Note that in accordance with (4.18) the fulfillment of the convolution and
the devolution in the binary representation (8.16) does not change the initial
number N, represented by the binary combination (8.16).
It is simple to fulfill convolutions and devolutions for the Fibonacci pcode
corresponding to the case p=1. In this case operations are carried out over three
Chapter 8
Fibonacci Computers
435
adjacent digits, namely over the lth , (l1)th and (l2)th digits. Let us consider
the fulfillment of these operations for the Fibonacci 1code (“Zeckendorf sum”):
(a) Convolution
0 1 1 1 1
7 = 1 0 0 1 1
1 0 1 0 0
(8.19)
(b) Devolution
1 0 0 0 0
5 = 0 1 1 0 0
0 1 0 1 1
(8.20)
The procedure that consists in the fulfillment of all possible convolutions or
devolutions in the initial binary combination (8.16) is named Code Convolution
and Code Devolution, respectively. It is easy to prove that the fulfillment of the
code convolution and the code devolution result in the socalled Convolute and
Devolute binary representations of the number N.
For the case p=1, the convolute and devolute binary representations of the
number N have peculiar indications. In particular, in the convolute binary rep
resentation, two binary numerals of 1 together do not meet and in the devo
lute binary representation two binary numerals of 0 together do not meet since
the highest 1 of the binary representation (8.16).
The rule of the “convolutiondevolution inversion” is of great importance
for technical applications. This rule consists in the following: the “convolution”
of the initial binary combination is equivalent to the “devolution” of the inverse
binary combination, and conversely. By using this rule, we can fulfill the reduc
tion to the “convolute” form for the example (8.19) as follows:
(a) Inversion of the initial binary combination 0 1 1 1 1:
01111 = 10000
(b) “Code devolution” of the inverse binary combination:
10000 = 01100 = 01011
(c) Inversion of the obtained code combination:
01011 = 10100
Now, let us consider the peculiarities of the “convolution” and “devolution”
for the lowest digits of the Fibonacci 1code. As is wellknown in the case p=1,
the weights of the two lowest digits of the Fibonacci 1code equal 1, that is,
F1=F2=1. And then the operations of the “devolution” and “convolution” for these
digits are fulfilled as follows:
10=01 (“devolution”) and 01=10 (“convolution”).
Alexey Stakhov
THE MATHEMATICS OF HARMONY
436
8.4.2. Radix of the Fibonacci pCode
What is the radix of the pFibonacci code? For the case p=0, the radix of the
binary system (8.10) is calculated as the ratio of the adjacent digit weights, that
is, 2k/2k1=2.
Apply this principle to the Fibonacci pcode (8.9) and examine the ratio
(8.21)
Fp(k)/Fp(k1).
The radix of the Fibonacci pcode (8.9) is the limit of the ratio (8.21). From
Chapter 4 we know that
F (k)
= τp,
lim p
k→∞ F (k − 1)
p
where τp is the golden pproportion.
This means that the radix of the Fibonacci pcode (8.9) is an irrational num
ber τp, the positive root of the characteristic equation xp+1xp1=0.
8.4.3. A Minimal Form of the Fibonacci pCode
The following theorem is of great importance for the theory of Fibonacci
pcodes.
Theorem 8.3. For the given integers p≥0 and n≥p+1, the arbitrary integer N
can be represented in the only form:
N=Fp(n)+R1,
where
(8.22)
0≤R1<Fp(np).
(8.23)
Proof. Note that the sequence of the Fibonacci pnumbers that are generat
ed by the recursive relation (4.18) at the seeds (4.19) is strictly an increasing
sequence starting with n=p+1. Consider the following number sequence:
(8.24)
Fp(p+1), Fp(p+2),…, Fp(n), Fp(n+1),… .
We can choose in this sequence the pair of adjacent Fibonacci pnumbers
Fp(n) and Fp(n+1) so that
(8.25)
Fp(n)≤N< Fp(n+1).
By subtracting the Fibonacci pnumber Fp(n) from all terms of the inequal
ity (8.25), we obtain:
0≤NFp(n)<Fp(n+1)Fp(n).
(8.26)
The formulas (8.22) and (8.23) follow directly from (8.26) because
Fp(n+1)Fp(n)=Fp(np).
Chapter 8
Fibonacci Computers
437
The theorem is proved.
Note that for the case p=0 we have: F0(n)=2n1 and therefore the expres
sions (8.22) and (8.23) take the following wellknown (in “binary” arith
metic) form:
(8.27)
N=2n1 +R1, 0≤R1<2n1.
Let us represent the integer N according to the formula (8.22) and
then let us represent all the remainders R1, R 2, Rk , which can arise in the
process of such representation, according to one and the same formula
(8.22). We will continue this process up to obtaining the remainder equal
to 0. As a result of this decomposition of the number N, we obtain a pecu
liar representation of integer N in the Fibonacci pcode (8.9). Its peculiarity
is that in the binary representation of the integer N given by (8.16) no less
than p binary numerals 0 follow after every digit al=1 from the left to the
right, that is,
(8.28)
al1= al2= alp=0.
Such representation of the integer N is called the Minimal Form or Minimal
Representation of the integer N in the Fibonacci pcode. This name reflects the
fact that for the case p=1 the minimal form of the integer N has a minimal num
ber of the binary numerals 1 in the binary representation of the Fibonacci 1
code among all binary representations of the same integer N.
For example, by using the above algorithm, we can obtain the following
minimal forms of the number 25 in the Fibonacci 1 and 2codes (Table 8.4).
Table 8.4. Minimal forms of the Fibonacci1 and 2codes
p = 1 F1 (i ) 55 34 21 13 8 5 3
25 = 0 0 1 0 0 0 1
p = 2 F2 (i ) 19 13 9 6 4 3 2
25 = 1 0 0 1 0 0 0
2 1 1
0 1 0
1 1 1
0 0 0
A peculiarity of the binary representations of number 25 given by Table
8.4 consists in the following. For the case p=1, not less than one binary nu
meral 0 follows after every binary numeral 1 from left to right in the binary
representation of number 25; for the case p=2 not less than two binary numerals
0 follow after every binary numeral 1 from left to right in the code representa
tion of the same number 25.
Corollary from Theorem 8.3. For a given p (p=0,1,2,3,…) every integer N
has the only minimal form in the Fibonacci pcode.
This means that there is a onetoone mapping of natural numbers onto
the minimal forms of the Fibonacci pcode (8.9).
Alexey Stakhov
THE MATHEMATICS OF HARMONY
438
Note that for the case p=0 (the classical binary code) every integer N has the
only representation in the form (8.10). This means that every code representa
tion (8.10) is its “minimal form.”
Now, let us prove the following theorem.
Theorem 8.4. For a given integer p≥0 by using the ndigit Fibonacci p
code in the minimal form we can represent Fp(n+1) integers in the range from
0 to Fp(n+1)1, inclusively.
Proof. To prove the theorem, recall that there is a onetoone map
ping between natural numbers and their minimal forms in the Fibonacci
pcode (8.9). Suppose that the number of different ndigit minimal forms
of the Fibonacci pcode (8.9) is equal to T p(n). Let us find a recursive
relation for Tp(n). First of all, we can write the following evidence equal
ities:
Tp(1)=Tp(2)=…=Tp(p)=1.
(8.29)
It is easy to prove that for the case n=p+1 we can represent only two num
bers in the minimal form, 0 and 1, as follows:
0 = 00
...0 ; 1 = 100
...0 .
N
N
p +1
p
This means that
Tp(p+2)=2.
(8.30)
Let n>p+1. Suppose that for this case the set A of ndigit minimal forms in
the Fibonacci pcode (8.9) consists of Tp(n) elements, that is, we can represent
Tp(n) different numbers in the minimal form. To obtain a recursive relation for
Tp(n), we divide the set A into two noncrossing subsets A0 and A1. All minimal
ndigit forms of the subset A0 begin from 0 and all minimal ndigit forms of the
subset A1 begin from 1. By taking away the first binary numeral 0 at the begin
ning of all minimal forms of the subset A0, we obtain the subset of (n1)digit
minimal forms. According to the inductive hypothesis, the number of elements
in it is equal to Tp(n1).
Next let us examine the subset A1. In accordance with the condition for
minimal form, the p binary numerals 0 follow after the binary numeral 1. This
means that all binary combinations of the subset A1 begin from the code combi
nation 100…0 (p+1 digits). By taking away these (p+1) digits standing at the
beginning of all minimal forms of the subset A1, we obtain the set of all (np1)
digit minimal forms. According to the inductive hypothesis, the number of ele
ments of this subset is equal to Tp(np1).
The next recursive relation for Tp(n) follows from this consideration:
Tp(n)= Tp(n1)+Tp(np1).
(8.31)
Chapter 8
Fibonacci Computers
439
If we compare the recursive relation (8.31) that is given at the seeds (8.29) and
(8.30) with the recursive relation (4.18) at the seeds (4.19), we can conclude:
Tp(n)=Fp(n+1).
The theorem is proved.
For the case p=1, the minimal form has a very simple indicator: in the mini
mal form two binary numerals 1 together do not meet. But the “convolute” form,
considered above, has the same property. This means that for the Fibonacci 1
code the “convolute” form coincides with the minimal form and the reduction of
the Fibonacci 1code combination to the minimal form can be performed by us
ing “convolutions.” The example (8.19) demonstrates the process of reduction of
the Fibonacci 1code combination to the minimal form. Also the notion of the
maximal form is very important for the Fibonacci 1code. The maximal form can
be obtained from the initial binary combination by means of “devolutions” and
it coincides with the “devolute” form. The example (8.20) demonstrates the pro
cess of reduction of the Fibonacci 1code combination from the minimal form of
the number 5 to its maximal form. Note that the operations of the reduction of
the Fibonacci 1code combinations to the minimal and maximal forms are the
most important operations of Fibonacci arithmetic.
8.4.4. Comparison of Numbers in the Fibonacci pCodes
Now, let us deduce a rule for the number comparison of Fibonacci pcodes.
With this purpose we compare the ndigit numbers corresponding to the sub
sets A0 and A1. As all numbers of the subset A1 have the binary numeral 1 in the
highest, that is, the nth digit, the minimal number of the subset A1 is equal to
Fp(n), the weight of the nth digit of the Fibonacci pcode (8.9). On the other
hand, all numbers of the subset A0 have the binary numeral 0 in the highest, that
is, the nth digit. Then according to Theorem 8.4, the maximal number, which
may be represented in the minimal form by using (n1) digits is equal to Fp(n)
1. Hence, the minimal number of the subset A1 is always greater than the maxi
mal number of the subset A0.
From this examination, the simple rule of comparison follows. The compar
ison of two numbers A and B represented in the minimal form of the Fibonacci
pcode (8.9) can be carried out digitbydigit starting with the highest digit un
til the first pair of noncoincident digits of the comparable codes A and B is found.
If the number A has the binary numeral 1 and the number B has the binary
numeral 0 in the first pair of the noncoincident digits, we can then conclude
that A>B. If the code combinations of the comparable numbers coincide for all
digits, then the numbers are equal to each other.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
440
Thus, the comparison of numbers in the Fibonacci pcodes is carried out in a
manner similar to the classical binary code, provided before the comparison we
reduce the comparable codes to their minimal form. This property (simplicity of
number comparison) is one of the important arithmetical advantages of the Fi
bonacci pcodes (8.9).
For example, we need to compare two numbers A=00111101101 and
B=00111110110 that are represented in the Fibonacci 1code. The number
comparison is carried out in two steps:
1. Reduce the comparable codes to their minimal form:
A = 00111101101 = 0100111001 = 0101001010
B = 00111110110 = 01001111000 = 01010011000 = 01010100000.
2. Compare digitbydigit the minimal forms of the number A and B begin
ning with the highest digit on the left:
A = 01010 [0]10010
B = 01010 [1] 00000.
We can see that the first noncoincident pair of the comparable combina
tions (from left to right) contains the binary numeral 0 in the minimal form of
the first number A and the binary numeral 1 in the minimal form of the second
number B. This means that B>A.
8.4.5. Redundancy of the Fibonacci pCodes
For the case p=0, the Fibonacci 0code (classical binary code) is non
redundant. However, for p>0, all Fibonacci pcodes (8.9) are redundant. And
their redundancy shows in the Multiplicity of the Fibonacci pcode represen
tation of one and the same integer N. Theorems 8.2 and 8.4 allow one to
calculate the redundancy of the Fibonacci pcodes for p>0 in comparison
with the classical binary code (p=0).
We can calculate the relative code redundancy r by the following
formula [177]:
n−m n
r=
= −1,
(8.32)
m
m
where n and m are the code length of the redundant and nonredundant codes,
respectively, for the representation of one and the same number range.
Note that the redundancy definition given by (8.32) characterizes a
relative increase of the code length of the redundant code relative to the non
redundant code for the representation of one and the same number range.
Chapter 8
Fibonacci Computers
441
Now, let us examine the range of number representation for the ndigit Fi
bonacci pcodes (p>0). The problem has two solutions. It follows from Theorem
8.2 that we can represent Fp(n+p+1) integers by using the ndigit Fibonacci p
code (8.9) if we represent integers N≥0 without the restrictions on the form of
the code representation. However, it follows from Theorem 8.4 that we can rep
resent Fp(n+1) integers N by using the ndigit Fibonacci pcode (8.9) if we use
only the minimal forms for the number representation.
For the code representation of the numbers that are in the number range
Fp(n+p+1) or Fp(n+1), it is necessary to use either m1≈log2Fp(n+p+1) or
m2≈log2Fp(n+1) binary digits of the nonredundant code, respectively. Using
(8.32) we obtain the following formulas for the calculation of the relative code
redundancy of the Fibonacci pcode (8.9):
n
r1 =
−1
(8.33)
log 2 Fp (n + p + 1)
r2 =
n
− 1.
log 2 Fp (n + 1)
(8.34)
The simplest redundant Fibonacci pcode is the code corresponding to the
case p=1. Let us calculate the limiting value of the relative redundancy for this
code. The formulas (8.33) and (8.34) for the Fibonacci 1code take the following
forms, respectively:
n
r1 =
−1
(8.35)
log 2 Fn +2
r2 =
n
− 1,
log 2 Fn +1
(8.36)
where Fn+1, Fn+2 are the classical Fibonacci numbers.
We can represent the Fibonacci numbers Fn+1, Fn+2 by using the Binet for
mulas (2.68). For any large n we can write the Binet formulas (2.68) in the fol
lowing approximate form:
τn
Fn ≈
.
(8.37)
5
Using (8.37) and substituting the approximate values for Fn+2 and Fn+1 into
the formulas (8.35) and (8.36), we obtain the following formulas:
n
−1
r1 =
(8.38)
(n + 2)log 2 τ − log 2 5
r2 =
n
− 1.
(n + 1)log 2 τ − log 2 5
(8.39)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
442
If we direct n to infinity in the expressions (8.38) and (8.39), we find that
they coincide for this case. Here, the utmost value of the relative redundancy
for the Fibonacci 1code is determined by the following expression:
1
r=
− 1 = 0.44.
log 2 τ
Thus, the utmost value of the redundancy of the Fibonacci 1code (Zecken
dorf representation) is a constant value equal to 0.44 (44%).
8.4.6. Surprising Analogies Between the Fibonacci and Genetic Codes
Among the biological concepts that have a level of general scientific signifi
cance and are well formalized, the genetic code plays a special role. The discov
ery of the now wellknown fact of the striking simplicity of the basic principles
of the genetic code is one of the major modern discoveries of modern science.
This simplicity consists of the fact that the inheritable information is encoded
by the texts of the threealphabetic words Triplets or Codonums compounded
on the base of the alphabet that consists of the four characters nitrogen bases:
A (adenine), C (cytosine), G (guanine), T (thiamine). The given recording sys
tem is unique for all boundless set of miscellaneous living organisms and is called
the Genetic Code [141].
It is known [141] that by using the threealphabetic triplets or codonums,
we can encode 21 items that include 20 amino acids and one more item called the
Stopcodonum (sign of the punctuation). It is clear that 43=64 different combi
nations (from four by three nitrogen bases) are used for encoding 21 items. In
this connection some of the 21 items are encoded by several triplets. It is called
the Degeneracy of the Genetic Code. Finding the conformity between triplets
and amino acids (or signs of the punctuation) is interpreted as Decryption of the
Genetic Code. Now, let us examine the 6digit Fibonacci 1code (Zeckendorf
representation) (8.1) that uses 6 Fibonacci numbers 1, 1, 2, 3, 5, 8 as digit weights:
(8.40)
N = a6 × 8 + a5 × 5 + a4 × 3 + a3 × 2 + a2 × 1 + a1 × 1.
There are the following surprising analogies between the 6digit Fibonacci
code and the genetic code:
1. The first analogy. For the representation of numbers the 6digit binary
Fibonacci code uses 26=64 binary combinations from 000000 up to 111111
which coincide with the number of triplets of the genetic code (43=64).
2. The second analogy. By using the 6digit Fibonacci code (8.40), we can
represent 21 integers including the minimal number 0 that is encoded by
the 6digit binary combination 000000 and the maximal number 20 that is
encoded by the 6digit binary combination 111111. Note that by using the
Chapter 8
Fibonacci Computers
443
triplet coding we can also represent 21 items including 20 amino acids and
one additional object, the stopcodonum or the sign of the punctuation that
indicates a termination of protein synthesis.
3. The third analogy. The main feature of the Fibonacci code is the Multiplic
ity of number representation. Except for the minimal number 0 and the max
imal number 20 that have the only code representations 000000 and 111111,
respectively, all the rest of numbers from 1 up to 19 have multiple representa
tions in the Fibonacci code, that is, they use no less than two code combina
tions for their representation. It is necessary to note that the genetic code
has the similar property called the Degeneracy of the genetic code.
Thus, between the Fibonacci code (8.40) and the genetic code based on
the triplet representation of amino acids, there are very interesting analogies
that allow us to consider the Fibonacci code to be a peculiar class of redun
dant codes among various ways of redundant coding. Thus we suggest that
the study of the Fibonacci code may be of great interest for the genetic code.
It is very possible that the similar analogies can become rather useful in the
designing of DNAbased biocomputers.
8.5. Fibonacci Arithmetic: The Classical Approach
8.5.1. Fibonacci Addition
Let us begin with developing the Fibonacci arithmetic. There are two meth
ods for its development. The first way is to use an analogy between classical bi
nary arithmetic and Fibonacci arithmetic (a “classical approach”). The second
way is to develop the “original Fibonacci arithmetic” based on some peculiari
ties of the Fibonacci pcodes.
Let us first start the development with the classical approach. As the
notion of the Fibonacci pcode given by (8.9) is a generalization of the notion
of the classical binary code (8.10), we can use the following method for
obtaining the arithmetical rules of the Fibonacci parithmetic. We begin by an
alyzing the corresponding rule for the classical binary arithmetic and then by
analogy we find a similar rule for the Fibonacci parithmetic.
We start from the addition rule for the Fibonacci pcode (8.9). It is well
known that the classical binary addition is based on the following identity for
the binary numbers:
2 k1+2k1=2k,
(8.41)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
444
where 2k1 and 2k are the weights of the (k1)th and kth digits of the binary code
(8.10), respectively.
To obtain the rule of number addition for the Fibonacci pcode, we ana
lyze the following sum:
(8.42)
Fp(k)+Fp(k),
where Fp(k) is the weight of the kth digit of the Fibonacci pcode (8.9).
Let p=1. For this case we have
(8.43)
Fp(k)=Fk,
where Fk is the classical Fibonacci number given by the recursive relation (8.2)
at the seeds (8.3). If we take into consideration (8.2) and (8.3), we can represent
the sum (8.42) as follows:
Fk+Fk=Fk+Fk1+Fk2
(8.44)
Fk+Fk=Fk+1+Fk2.
(8.45)
The following addition rule of two binary digits ak + bk with the same index k
for the Fibonacci 1code follows from (8.44) and (8.45) represented in Table 8.5.
The following rules of the Fibonacci 1addition of the binary numerals of
the kth digits ak+bk follow from
Table 8.5. A rule of summation
Table
8.5:
for the Fibonacci 1code
Rule 8.1. At the addition of
ak + bk = ck +1 sk ck −1 ck −2
the binary numerals 1+1 of the k
0 + 0 = 0
0
0
0
th digits the carryover of the bi
0 + 1 = 0
1
0
0
nary numeral 1 from the kth dig
1 + 0 = 0
1
0
0
it to the next two digits arises.
Rule 8.2. There are two
1 + 1 = 0
1
1
1 (a)
0
0
1 (b) ways of carryingover the for
1 + 1 = 1
mation. In method (a) (see Ta
ble 8.5) the binary numeral 1 is attributed to the kth digit of the intermediate
sum sk and the carryover of the binary numeral 1 to the next two lower digits,
that is, to the (k1)th and (k2)th digits, ck1 and ck2, occurs. Method (b) (see
Table 8.5) assumes another rule of the Fibonacci 1summation of binary dig
its. The binary numeral 0 is attributed to the kth digit of the intermediate
sum sk and the binary numeral 1 is carriedover to the next two digits, that is,
to the (k+1)th and (k2)th digits, ck+1 and ck2, appear.
The summation of the multidigit numbers in the Fibonacci 1code is
fulfilled in accordance with the Fibonacci 1summation according to Table
8.5. However, we should adopt the following rules:
Rule 8.3. Before summation, the addends are reduced to the minimal
form.
Chapter 8
Fibonacci Computers
445
Rule 8.4. In accordance with Table 8.5, it is necessary to form the multi
digit intermediate sum and multidigit carryover.
Rule 8.5. The multidigit intermediate sum is reduced to the minimal
form and is then summarized with the multidigit carryover.
Rule 8.6. The addition process continues in accordance with rules 8.4 and
8.5 until the multidigit carryover equal to 0 is obtained. The last intermediate
sum, which is reduced to the minimal form, is the result of the addition.
For the Fibonacci 1addition it is necessary to fulfill the additional rule 8.7.
Rule 8.7. Let us examine the case, where we have two binary numerals 1 in
the kth digits of the addends. It follows from the property of the minimal form
that the binary numerals of the (k+1)th and (k1)th digits of both of the
addends are always equal to 0. It is clear that for this case the intermediate sum
that appears at the addition of the (k+1)th and (k1)th digits of both of the
addends are also always equal to 0. This means that we can place one of the
carryovers that appear at the addition of the kth significant digits (1+1) at
once to the (k1)th digit of the intermediate sum (for the (a)method) or to
the (k+1)th digit of the intermediate sum (for the (b)method).
We can demonstrate the above Fibonacci addition rules in the example below.
Example 8.1. Summarize the numbers 31=10011011 and 22=01011010
represented in the Fibonacci 1code.
The first step is to reduce the binary representations of the numbers 31 and
22 to the minimal form:
31=10100100; 22=10000010.
The second step is the formation of the multidigit intermediate sum S1 and
multidigit carryover C1 in accordance with method (a) of Table 8.5:
31 = 10100100
+
22 = 10000010
___________
S1 = 11100110
C1 = 00100000
The third step is to reduce the intermediate sum S1 to the minimal form:
S1=11100110=100101000.
The fourth step is the addition of S1 and C1:
S1 = 100101000
+
C1 = 000100000
____________
S2 = 100111000
C2 = 000001000
Alexey Stakhov
THE MATHEMATICS OF HARMONY
446
The fifth step is to reduce the intermediate sum S2 to the minimal form:
S2=100111000=101001000.
The sixth step is the addition of S2 and C2:
S2 = 101001000
+
C2 = 000001000
____________
S3 = 101001100
C3 = 000000010
The seventh step is to reduce the intermediate sum S3 to the minimal form:
S3=101001100=101010000.
The eighth step is the addition of S3 and C3:
S3 = 101010000
+
C3 = 000000010
____________
S4 = 101010010
C4 = 000000000
The addition is over because the carryover C4=0 appears in the eighth step.
8.5.2. Direct Fibonacci Subtraction
The wellknown “direct” method of number subtraction in classical bina
ry arithmetic (p=0) is based on the following property of binary numbers:
(8.46)
2n+k2n=2n+k1+2n+k2+…+2n .
Now, let us write a similar identity for the Fibonacci 1numbers:
(8.47)
Fn+k Fn= Fn+k2+Fn+k3+…+Fn1 .
Using the identity (8.47) and the Fibonacci recursive relation (8.2), we
obtain the following Fibonacci 1subtraction table represented in Table 8.6.
The reader of the present book can “deduce” the rules of
Table 8.6. The
the
“direct” Fibonacci psubtraction by using the recursive
rule of subtrac
relation for the Fibonacci pnumbers.
tion for the
Fibonacci 1code
The “direct” Fibonacci 1subtraction of multidigit num
0−0=0
bers is based on the following rules:
1−1=0
Rule 8.8. The numbers are reduced to the minimal form
1 − 0 = 0 1 1 before subtraction.
10−1=01
Rule 8.9. After reduction to the minimal form, the numbers
100−1=11
are compared by their value and then in accordance with Table
1 0 0 0 − 1 = 1 1 1 8.6, the smaller number being subtracted from the larger one.
Chapter 8
Fibonacci Computers
447
8.5.3. Fibonacci Inverse and Additional Codes
It is wellknown that the classical binary arithmetic employs the notions of
“inverse” and “additional” codes for the representation of negative numbers. This
approach allows one to reduce the binary subtraction to binary addition. For
the case of integer representation, the following mathematical properties under
lie the “inverse” code (Ginv) and the “additional” code (Gad):
Ginv=2n1 |G|
(8.48)
Gad=2n |G|,
(8.49)
where |G| is the module of the initial integer G.
Note that the “” (“minus”) sign is encoded by the binary numeral 1 and
the “+” (“plus”) sign by the binary numeral 0. They are placed at the begin
ning of the code combination. The digital part coincides with its direct code
for the positive number G, and is determined by (8.48) and (8.49) for the
negative number G.
The notions of inverse and additional codes have the following generaliza
tion for the Fibonacci 1code. By analogy to (8.48) and (8.49) we can introduce
the notions of the “Fibonacci inverse” and “Fibonacci additional” ndigit 1codes:
Ginv=Fn+11 |G|
(8.50)
Gad=Fn+1 |G|,
(8.51)
where Fn+1 is the Fibonacci 1number, and G is any integer.
The expressions (8.50) and (8.51) have the following code interpretation.
Suppose that the initial integer G is represented in the minimal form of the n
digit Fibonacci 1code (8.1). It follows from Theorem 8.4 that the integer G is in
the following number range:
(8.52)
0 ≤ G ≤ Fn −1.
On the other hand, the following important expressions that connect the
Fibonacci inverse and additional codes follow from (8.50) and (8.51):
Ginv + G = Fn −1 − 1
(8.53)
(8.54)
Gad = Ginv + 1.
Using the general property (2.20), we can represent the number Fn+1–1 as
follows:
Fn+1–1= Fn1+Fn2+…+F2+F1.
Note that the sum (8.55) has the following code interpretation:
Fn +1 − 1 = 011
...1 = Ginv + G .
N
n −1
(8.55)
(8.56)
Alexey Stakhov
THE MATHEMATICS OF HARMONY
448
The formula (8.56) can be used for the introduction of the following algo
rithm for obtaining the Fibonacci inverse and Fibonacci additional 1codes
from the initial code G (for example, from the initial Fibonacci 1code
G=10010100):
1. By using the operation of “devolution” for the initial code G:
G=10010100=01110011=01101111.
2. By using the operation of “inversion” for all binary digits of the “devolute”
form of the number G, except for the higher digit:
01101111 = 00010000.
The resulting binary combination is the Fibonacci inverse 1code of G:
Ginv=00010000.
3. By adding the binary numeral 1 to the lowest digit of the Fibonacci in
verse 1code in accordance to (8.54) and by reducing the obtained code com
bination to the minimal form, we obtain the Fibonacci additional 1code G:
Gad=00010001=00010010.
Note that in the general case p≥0, the expressions for the Fibonacci inverse
and additional pcodes have the following forms:
Ginv = Fp ( n + 1) − 1 − G
(8.57)
Gad = Fp ( n + 1) − G .
(8.58)
We can see that the expressions (8.48), (8.49) and (8.53), (8.54) are partial
cases of expressions (8.57) and (8.58) because for the case p=0, F0(n+1)=2n
and for the case p=1, F1(n+1)=Fn+1.
Now, let us introduce the following rule of the sign encoding for the Fibonacci
inverse and additional 1codes. We will be encoding the “−” sign with the binary
numeral 1 and the “+” sign with the binary numeral 0. The sign of the code com
binations is put at the beginning of the numerical part of the code. Hence, the
Fibonacci inverse and additional 1codes of the negative number
G=10010100
have the following forms, respectively:
Ginv=1.00010000
Ginv=1.00010010.
Note that the numerical part of the Fibonacci inverse and additional 1
codes of the positive number G, coincide with the Fibonacci 1code of the
same number G. For example, the Fibonacci inverse and additional 1codes of
the positive number G=+10010100 have the following form:
Gad=0.10010100.
Chapter 8
Fibonacci Computers
449
Let us formulate the Fibonacci subtraction algorithm based on the notions
of the Fibonacci inverse and additional 1codes:
1. Represent the initial numbers in the Fibonacci inverse or additional 1
codes.
2. Sum the numerical parts in accordance with the rule of the Fibonacci 1
addition and the signs in accordance with the rule of classical binary addi
tion. The addition result is represented in the Fibonacci inverse or addition
al 1codes. If the sign of the code combination is equal to 1, it means that the
addition result is a negative number. For obtaining the absolute value of the
addition result, it is necessary to use the above algorithm of code transfor
mation for the Fibonacci inverse or additional 1codes for the numerical part
of the addition result.
8.6. Fibonacci Arithmetic: An Original Approach
8.6.1. Basic Micro Operations
As mentioned above the multiplicity of number representation is the
main peculiarity of the Fibonacci pcode (8.9) for the case p>0 in compar
ison to the classical binary code (8.10). By using the above operations of
“devolution” and “convolution”, we can change the form of representation
of one and the same number. This means that the binary 1’s can move in the
code combination to the left or to the right. This fact allows us to develop
an original approach to the Fibonacci arithmetic based upon the socalled
Basic Microoperations.
We can introduce the following four “basic microoperations” that are used
in the Fibonacci processor for fulfillment of the logical and arithmetical opera
tions: (a) Convolution; (b) Devolution; (c) Replacement; (d) Absorption.
Consider these microoperations for the case of the Fibonacci 1code
(Zeckendorf representation). We showed above that for the case p=1 the
Convolution and Devolution are the following code transformations that are
carried out for all the adjacent triple digits of one and the same Fibonacci 1
code combination.
Convolution:
[011→100]
Devolution:
[100→011]
Alexey Stakhov
THE MATHEMATICS OF HARMONY
450
The microoperation of Replacement
 1 0
↓ = 
 0 1


is a twoplaced microoperation that is fulfilled at the same digit of two reg
isters, the top register A and the bottom register B. Examine the case when
the register A has the numeral 1 in the kth digit and the register B has the
numeral 0 in the same digit. For this condition we can fulfill the following
microoperation. We shift the above binary numeral 1 of the top register A to
the bottom register B. This microoperation is named Replacement. Note
that this operation can be carried out only for the condition if the kth digits
of the registers A and B are equal to 1 and 0, respectively.
The microoperation of Absorption
 1 0
7 = 
 1 0
is a twoplaced microoperation that consists in mutually annihilating two 1’s
in the kth digit of two registers A and B and replacing them with the binary
numerals 0.
It is necessary to pay attention to the following “technical” peculiarity of the
above “basic microoperations.” At register interpretation of these microopera
tions, each microoperation may be considered to be the inversion of the triggers
(flipflops) that are involved in the microoperation. This means that each mi
crooperation is carried out by trigger switching.
8.6.2. Logical Operations
We can demonstrate the possibility of carrying out all logical operations by
using the above four “basic microoperations.”
Let us construct all the possible “replacements” from top register A to
bottom register B:
A = 1 0 0 1 0 0 1
↓
↓
B = 0 1 0 0 1 0 1
A′ = 0 0 0 0 0 0 1
B′ = 1 1 0 1 1 0 1
We obtain two new code combinations A′ and B′ as the outcome of the “re
placement.” We can see that the binary combination A′ is a logical “conjunction”
(∧) of the initial binary combinations A and B, that is,
Chapter 8
Fibonacci Computers
451
A′ = A ∧ B
and the binary combination B′ is a logical “disjunction” (∨) of the initial code
combinations A and B , that is,
B′ = A ∨ B .
The logical operation of “module 2 addition” is performed by means of the
simultaneous fulfillment of all possible microoperations of “replacement”
and “absorption.” For example,
A = 1 0 1 0 0 1 1 0 1
7
↓
↓ ↓ 7
B = 1 1 0 0 1 0 0 0 1
A′ = 0 0 0 0 0 0 0 0 0 = const 0
B′ = 0 1 1 0 1 1 1 0 0 = A ⊕ B
We can see that the two new code combinations A′ = const 0 and B′ = A ⊕ B
are the result of this code transformation. The logical operation of “code A inver
sion” is reduced to fulfillment of the operation of “absorption” at the initial code
combination A and the special binary combination B=const 1 (see below):
A = 1 0 1 0 0 1 1 0 1
7 7
7 7 7
B = 1 1 1 1 1 1 1 1 1
A′ = 0 0 0 0 0 0 0 0 0 = const 0
B′ = 0 1 0 1 1 0 0 1 0 =
A
8.6.3. The Counting and Subtracting of Binary 1
Let us demonstrate the possibility of carrying out the simplest arithmet
ical operations by using the “basic microoperations.” We start with the oper
ations of “counting” and “subtracting” of the binary numerals 1.
The “counting” of the binary numerals 1 in the Fibonacci 1code (a
“summing counter”) is carried out by using “convolution.” For example, the
transformation of the initial Fibonacci 1code combination 01010 the minimal
form of the number 4
Fi = 5 3 2 1 1
4 = 0 1 0 1 0
to the next Fibonacci code combination of the number 5 is carried out in the
“summing Fibonacci counter” according to the rule:
Alexey Stakhov
THE MATHEMATICS OF HARMONY
452
Fi = 5 3 2 1 1
4 = 0 1 0 1 0 + 1
5 = 0 1 0 1 1
5 = 0 1 1 0 0
5 = 1 0 0 0 0
Here, in the top row we see Fibonacci numbers 5, 3, 2, 1, 1 that are the
weights of the Fibonacci 1code. The second row is a representation of the
number 4 in the minimal form (01010). We can see that the binary numeral 1 is
added (+1) to the lowest digit of the Fibonacci binary combination 01010.
Then, the initial binary combination 4=01010 is transformed into the binary
combination 5=01011 that is the Fibonacci binary representation of the next
number 5 (the third row). After that the Fibonacci binary combination 01011
is reduced to the minimal form. This transformation is fulfilled in 2 steps by
using the “convolutions.” The first step is to carry out the “convolution” over
the three lowest digits of the binary combination 01011 (the third row). Using
the “convolution” [011→100], we transform the Fibonacci binary representa
tion of the number 5=01011 to another Fibonacci binary representation of the
same number 5=01100 (the fourth row). Then, we carry out the “convolution”
over the next group 011 of the Fibonacci binary combination 5=01100 (the
fourth row). In the fifth row we see the minimal form of the number 5=10000.
We can continue the “counting” of the binary numerals 1 as follows:
6 = 10000 + 1 = 10001 = 10010
7 = 10010 + 1 = 10011 = 10100.
If we add the binary numeral 1 to the lowest digit of the number 7=10100,
we obtain:
10100+1=10101=10110=11000=00000.
This situation is well known in computer engineering and is named “over
filling” of the “summing counter.”
The “subtracting” of the binary numerals 1 (a “subtracting counter”) is
fulfilled by using “devolution.” For this purpose the initial Fibonacci 1code
combinations are reduced to the “devolute” or “maximal” form, and then the
binary numeral 1 is subtracted sequentially from the lowest digit:
Fi = 5 3 2 1 1
5 = 1 0 0 0 0
5 = 0 1 1
5 = 0 1 0 1 1 − 1
4 = 0 1 0 1 0
Chapter 8
Fibonacci Computers
453
Here, in the second row we see the representation of the number 5 in the
minimal form (5=10000). In the third row we carry out the “devolution” over
the highest three digits [100→011] of the binary representation of the initial
binary combination 5=10000. In the fourth row we carry out the next “devolu
tion” over the next three digits of the binary representation of the number
5=01100 (see the third row). As the outcome we obtain the maximal form of the
number 5=01011 (the fourth row). Then, we subtract the binary numeral 1 from
the lowest digit of the binary combination 5=01011. A result of this transforma
tion 4=01010 is represented in the fifth row. Then, we can continue the “sub
tracting” of the 1’s as follows:
4 = 01010 = 01001 (" devolutions ")
3 = 4 − 1 = 01001 − 1 = 01000
3 = 01000 = 00110 = 00101 (" devolutions ")
2 = 3 − 1 = 00101 − 1 = 00100
2 = 00100 = 00011 (" deevolution ")
1 = 2 − 1 = 00011 − 1 = 00010
1 = 00010 = 00001 (" devolution ")
0 = 1 − 1 = 00001 − 1 = 00000.
8.6.4. Fibonacci Summation
The idea of summation of two numbers A and B is based on the “basic micro
operations.” The first step is to shift all binary numerals 1 from the top register A
to the bottom register B. With this purpose we use “replacement,” “devolution,”
and “convolution.” The sum is formed in register B.
For example, let us sum the following numbers
A0=010100100 and B0=001010100.
The first step of the Fibonacci addition consists in the “replacement” of
all possible binary numerals 1 from register A to register B:
A0 = 0 1 0 1 0 0 1 0 0
↓
↓
B0 = 0 0 1 0 1 0 1 0 0
A1 = 0 0 0 0 0 0 1 0 0
B1 = 0 1 1 1 1 0 1 0 0
The second step is the fulfillment of all possible “devolutions” in the binary
combination A1 and all possible “convolutions” in the binary combination B1:
Alexey Stakhov
THE MATHEMATICS OF HARMONY
454
A1=000000100→ A2=000000011
B1=011110100→ B2=100110100.
The third step is the “replacement” of all possible binary numerals 1 from
register A to register B:
A2 = 0 0 0 0 0 0 0 1 1
↓ ↓
B2 = 1 0 0 1 1 0 1 0 0
A3 = 0 0 0 0 0 0 0 0 0
B3 = 1 0 0 1 1 0 1 1 1
The addition is over because all binary numerals 1 are shifted from the regis
ter A to the register B. After reducing the binary combination B3 to the minimal
form, we obtain the sum B3=A+B, which is represented in the minimal form:
B3=100110111=101001001=101001010=A+B.
Thus, the addition is reduced to a sequential fulfillment of the microopera
tions of “replacement” over two binary combinations A and B and the micro
operations of both “convolution” over the binary combination B and “devolu
tion” over the binary combination A.
8.6.5. Fibonacci Subtraction
The idea of direct subtraction of number B from number A which is based on
the “basic microoperations,” consists of the mutual “absorption” of the binary
numerals 1 in the binary combinations of numbers A and B until one of them
becomes equal to 0. To perform this process, we have to sequentially carry out the
aforementioned microoperations of “absorption” and “devolution” over the code
combinations A and B. The subtraction result is always formed in the register
that contains the larger number. If the subtraction result is formed in the top
register A, it follows from this fact that the sign of the subtraction result is “+”
(“plus”); in the opposite case the subtraction result has the “−” (“minus”) sign.
Let us now demonstrate this idea for the following example. Subtract the
number B0=101010010 from the number A0=101001000 in the Fibonacci 1code.
The first step is “absorption” of all possible binary 1’s in the numbers A and B:
A0 = 1 0 1 0 0 1 0 0 0
7 7
B0 = 1 0 1 0 1 0 0 1 0
A1 = 0 0 0 0 0 1 0 0 0
B1 = 0 0 0 0 1 0 0 1 0
Chapter 8
Fibonacci Computers
455
The second step is “devolution” of the code combinations A1 and B1:
A1=000001000→ A 2=000000110
B1=000010010→ B2=000001101.
The third step is “absorption” of A2 and B2:
A2 = 0 0 0 0 0 0 1 1 0
7
B2 = 0 0 0 0 0 1 1 0 1
A3 = 0 0 0 0 0 0 0 1 0
B3 = 0 0 0 0 0 1 0 0 1
The fourth step is “devolution” of A3 and B3:
A3=000000010→ A4=000000001
B3=000001001→ B4=000000111.
The fifth step is “absorption” of A4 and B4:
A4 = 0 0 0 0 0 0 0 0 1
7
B4 = 0 0 0 0 0 0 1 1 1
A5 = 0 0 0 0 0 0 0 0 0
B5 = 0 0 0 0 0 0 1 1 0
The subtraction is finished. After reducing the code combination B5 to the
minimal form we obtain:
B5=000001000.
The subtraction result is in the register B. This means that the sign of the
subtraction result is “−” (“minus”), that is, the difference of the numbers AB is
equal to:
R=AB=000001000.
8.7. Fibonacci Multiplication and Division
8.7.1. The Egyptian Method of Multiplication
In order to find the algorithms for Fibonacci multiplication and division, we
use an analogy with classical binary multiplication and division. We start with
multiplication. To multiply two numbers A and B in the classical binary code
Alexey Stakhov
THE MATHEMATICS OF HARMONY
456
(8.10) we should represent the multiplier B in the form of the ndigit binary
code. The product P=A×B can then be written in the following form:
(8.59)
P=A×bn2n1+ A×bn12n2+…+ A×bi2i1+…+ A×b120,
where bi is the binary numeral of the multiplier B. It follows from (8.59) that
the binary multiplication is reduced to form the partial products of the kind
A×bi2i1 and their addition. The partial product A×bi2i1 is made up by shifting
the binary code of number A to the left by (i1) digits.
The binary multiplication algorithm based on (8.59) has a long history
and goes back in its origin to the Egyptian Doubling Method [169].
As is known, the Egyptian number system was decimal, but nonposition
al. The Egyptians represented the “nodal” numbers of their number system 1,
10, 100, 1000, 10 000, etc. by using special hieroglyphs. Then, in the record of
any number, for example, 325, they used the 5 hieroglyphs representing the
“nodal” number 1, the 2 hieroglyphs representing the “nodal” number 10, and
the 3 hieroglyphs representing the “nodal” number 100.
The invention of the Doubling Method that underlies the arithmetical oper
ations of multiplication and division is the main achievement of Egyptian arith
metic. In order to multiply the number 35 by the number
Table 8.7. Egyptian
12, the Egyptian mathematician acted as follows. He built
method of
multiplication
up a special table of numbers (see Table 8.7). The binary
numbers 2k (k=0,1,2,3,…) were placed in the first column
1 35
of Table 8.7. In the second column of Table 8.7 we can see
2 70
the numbers 35, 70, 140, 280, that is, the numbers formed
/4 140 → 140
from the multiplier 35 according to the Doubling Method.
/8 280 → 280
420
After that the Egyptian mathematician used the in
clined line to mark those binary numbers of the first
column whose sum is equal to the second multiplier (12=8+4). Then he
selected those numbers of the second column corresponding to the marked
“binary” numbers of the first column. The result of multiplication is equal
to the sum of the selected numbers of the second column (140+280=420).
The analysis of the Egyptian method of multiplication based on the Dou
bling Method results in a rather unexpected conclusion. The representation
of the multiplier 12 in the form 12=8+4 corresponds to its representation in
the binary system (12=1100). On the other hand, if we represent the second
multiplier 35 in the binary system (35=100011), then the “doubling” of the
number 35 corresponds to the displacement of the binary code of the multi
plier 35=100011 to the left (70=1000110, 140=10001100, etc.). In other
words, the Egyptian multiplication method based on the Doubling Method
coincides with the basic multiplication algorithm used in modern computers!
Chapter 8
Fibonacci Computers
457
This basic algorithm follows from the representation of the product in the form
of (8.56).
8.7.2. The Egyptian Method of Division
The Egyptians also used the Doubling Method for the division of numbers.
If, for example, we need to divide the number 30 by the number 6, the Egyp
tian mathematician acted as follows (see Table 8.8).
The first stage of the division consists of the following. Let us write in the
first column of Table 8.8 the binary numbers 2k (k=0, 1, 2, 3,…). Then, let us
write in the second column the divisor 6 and
Table 8.8. Egyptian method
its “doubling” numbers 12, 24, and 48. In ev
of division
ery “doubling” step we will compare the divi
6 ≤ 30
6≤6
/1
dend 30 with the numbers of the second col
2
12 ≤ 30
12 > 6
umn until the next “doubling” number (the
24 ≤ 30
/4
number 48 in this example) becomes more
8
48 > 30
than the dividend (48>30). After that we shall
4 + 1 = 5 30 − 24 = 6 6 − 6 = 0
subtract the “doubling” number of the previ
ous row (3024=6) from the dividend 30 and mark the corresponding binary
number of the first column (the number 4) with the inclined line. After that
we shall perform the same procedure with the remainder 6 (see the third
column) until the “doubling” number becomes strictly more than the differ
ence 6 (12>6). By marking the number 1 of the first column with the inclined
line and then by subtracting the divisor 6 from the remainder 6, we obtain the
number 0 (66=0). This means that the division is over. After that we can see
that the sum of the binary numbers of the first column marked by the inclined
line (4+1=5) is equal to the result of the division.
Thus, after due consideration one is astonished by the genius of the Egyp
tian mathematicians who several millennia ago invented the methods of mul
tiplication and division that are today used in our modern computers!
8.7.3. Fibonacci Multiplication
Analysis of the Egyptian Doubling Method allows us to develop the follow
ing method of “Fibonacci pmultiplication.”
Let us consider the product P=A×B, where the numbers A and B are rep
resented in the Fibonacci pcode (8.9). By using the representation of the
multiplier B in the Fibonacci pcode (8.9), we can write the product P=A×B
in the following form:
Alexey Stakhov
THE MATHEMATICS OF HARMONY
458
P=A×bnFp(n)+A×bn1Fp(n1)+…+A×bi Fp(i)+…+A×b1 Fp(1),
(8.60)
where Fp(i) is the Fibonacci pnumber.
Note that the expression (8.60) is a generalization of the expression
(8.59) that underlies the algorithm of the “binary” multiplication. The fol
lowing algorithm of the Fibonacci pmultiplication follows directly from
the expression (8.60). The multiplication is reduced to the addition of the
partial products of the kind A×biFp(i). They are formed from the multiplier
A according to the special procedure that reminds one of the Egyptian Dou
bling Method. Let us demonstrate the “Fibonacci multiplication” for the
case of the simplest Fibonacci 1code (p=1).
Example 8.2. Find the following product: 41×305.
Solution:
1. Construct a table consisting of
Table 8.9. Fibonacci multiplication
three columns marked by F, G and P
F
G
P
(see Table 8.9).
1
305
2. Insert the Fibonacci 1sequence
1
305
(the classical Fibonacci numbers) 1,
/2
610
→ 610
1, 2, 3, 5, 8, 13, 21, 34 into the Fcol
3
915
umn of Table 8.9.
/5
1525
→ 1525
3. Insert the generalized Fibonacci 1
8
2440
sequence: 305, 305, 610, 915, 1525,
13
3965
2440, 3965, 6505, 10370, which is built
21
6505
up from the first multiplier 305 ac
/34
10370 → 10370
cording to the “Fibonacci recursive
41 = 34 + 5 + 2 41×305 = 12505
relation” into the Gcolumn.
4. Mark by inclined line (/) and bold
type all numbers of the Fcolumn, which make up the second multiplier 41
in the sum (41=34+5+2).
5. Mark by bold type all Gnumbers corresponding to the marked Fnum
bers and rewrite them in the Pcolumn.
6. Summing all Pnumbers, we obtain the product: 41×305=12505.
The multiplication algorithm is easily generalized for the case of Fibonac
ci pcodes.
8.7.5. Fibonacci Division
We can apply the above Egyptian method of division to construct the
algorithm of Fibonacci pdivision. Consider this method for the following
specific example.
Chapter 8
Fibonacci Computers
459
Example 8.3. Divide the number 481 (the dividend) by the number 13 (the
divisor) in the Fibonacci 1code.
Solution has two steps as shown below.
8.7.5.1. The first stage
1. Construct the table, which consists of three columns marked by F, G
and D (see Table 8.10).
2. Insert the Fibonacci 1sequence (the classical Fibonacci numbers) 1, 1,
2, 3, 5, 8, 13, 21, 34, 55 to the Fcolumn of Table 8.10.
3. Insert the generalized Fibonacci 1se
Table 8.10. The first stage
of the Fibonacci division
quence: 13, 13, 26, 39, 65, 104, 169, 273,
F
G
D
442, 615 formed from the divisor 13 ac
≤
1
13
481
cording to the “Fibonacci recursive rela
≤
1
13
481
tion” into the Gcolumn.
≤
2
26
481
4. Compare sequentially every Gnumber
≤
3
39
481
with the dividend 481, inscribed into the
≤
5
65
481
Dcolumn, and fix the result of compari
≤
8
104
481
son (≤ or >) until we obtain the first com
≤
13
169
481
parison result of the kind: 615>481.
≤
481
21 273
5. Mark with the inclined line (/) and bold
/34 442
≤
481
type the Fibonacci number 34, which cor
55 615
>
481
responds to the preceding Gnumber 442,
R1
= 481− 442 = 39
and mark the latter with bold type.
6. Calculate the difference: R 1=481
442=39.
Table 8.11. The second
stage of the Fibonacci
division
F
1
1
2
/3
5
R2
G
D
13
39
≤
13
39
≤
26
39
≤
39
39
≤
65
>
39
= 39 − 39 = 0
8.7.5.2. The second stage
The second stage of the Fibonacci 1division is
a repetition of the first stage when we instead use
the dividend 481, the remainder R1=39 (see Table
8.11).
The second remainder R 2=39–39=0, which
means the Fibonacci 1division is complete. The
result of the division is equal to the sum of all the
marked Fnumbers obtained throughout all stages
(see Tables 8.10 and 8.11), that is, 34+3=37.
Alexey Stakhov
THE MATHEMATICS OF HARMONY
460
8.8. Hardware Realization of the Fibonacci Processor
8.8.1. A Device for Reduction of the Fibonacci Code to Minimal Form
The devices for the “convolution” and “devolution” play an important
role in the technical realization of arithmetical operations in the Fibonacci
code. They can be designed on the basis of the binary register, which has
special logical circuits for the “convolutions” and “devolutions.” Each digit of
the register contains a binary “flipflop” (trigger) and logical elements. The
operations of “convolution” [011→100] and “devolution” [100→011] can be
carried out by means of the inversion of the “flipflops” (triggers).
One of the possible variants of the “convolution” register or “the device
for the reduction of the Fibonacci code to the minimal form” is shown in Fig.
8.1. This device consists of the five RStriggers and the logical elements
AND, OR which are used to carry out the “convolution.” The “convolution” is
carried out by means of the logical elements AND1 AND5 and OR which stand
before the R and Sinputs of the triggers. The logical element AND1 fulfills
the “convolution” of the 1st digit to the 2nd digit. Its two inputs are connect
ed with the direct output of trigger T1 and the inverse output of trigger T2.
The 3rd input is connected with the synchronization input C. The logical
element AND1 analyzes the states Q1 and Q2 of the triggers T1 and T2. If Q1=1
and Q2=0, it means that the convolution conditio